Sequences and Series Exercises: Limits & Formulas

CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 SEQUENCES " 2 1 3 2 1 4 3 1. a" œ 11#1 œ 0, a# œ " ## œ 4 , a$ œ 3# œ 9 , a% œ 4# œ 16 1 2. a" œ 1!1 œ 1, a# œ #"! œ "2 , a$ œ 3!1 œ 16 , a% œ 4!1 œ 24 3. # $ % & ) (1) (1) " " " a" œ (#1)1 œ 1, a# œ (4" 1 œ 3 , a$ œ 6 1 œ 5 , a% œ 8 1 œ 7 4. a" œ 2 (1)" œ 1, a# œ 2 (1)# œ 3, a$ œ 2 (1)$ œ 1, a% œ 2 (1)% œ 3 # $ % 5. a" œ #2# œ "# , a# œ 22$ œ "# , a$ œ #2% œ "# , a% œ 22& œ "# # $ % 15 6. a" œ 2 # " œ "# , a# œ 2 2# 1 œ 43 , a$ œ 2 2$ 1 œ 87 , a% œ 2 2% " œ 16 " 15 31 63 7. a" œ 1, a# œ 1 "# œ 3# , a$ œ 3# #"# œ 74 , a% œ 74 #"$ œ 15 8 , a& œ 8 #% œ 16 , a' œ 32 , 255 511 1023 a( œ 127 64 , a) œ 128 , a* œ 256 , a"! œ 512 ˆ #" ‰ ˆ "6 ‰ " 3 œ 6 , a% œ 4 " " a* œ 362,880 , a"! œ 3,628,800 8. a" œ 1, a# œ "# , a$ œ œ #"4 , a& œ ˆ #"4 ‰ 5 " " œ 1#"0 , a' œ 7#"0 , a( œ 5040 , a) œ 40,320 , 9. a" œ 2, a# œ (1)# (2) œ 1, a$ œ (1)2 (1) œ "# , a% œ (1)% ˆ "# ‰ (1)& ˆ 4" ‰ œ 4" , a& œ œ 8" , # # 10. a" œ 2, a# œ 1†(#2) œ 1, a$ œ 2†(31) œ 32 , a% œ 3†ˆ 23 ‰ 4†ˆ " ‰ œ "# , a& œ 5 # œ 52 , a' œ 3" , 4 # $ " " " a' œ 16 , a( œ 3"# , a) œ 64 , a* œ 1#"8 , a"! œ 256 a( œ 27 , a) œ "4 , a* œ 29 , a"! œ "5 11. a" œ 1, a# œ 1, a$ œ 1 1 œ 2, a% œ 2 1 œ 3, a& œ 3 2 œ 5, a' œ 8, a( œ 13, a) œ 21, a* œ 34, a"! œ 55 12. a" œ 2, a# œ 1, a$ œ "# , a% œ ˆ "# ‰ 1 ˆ"‰ œ #" , a& œ ˆ# " ‰ œ 1, a' œ 2, a( œ 2, a) œ 1, a* œ "# , a"! œ "# # 13. an œ (1)n1 , n œ 1, 2, á 14. an œ (1)n , n œ 1, 2, á 15. an œ (1)n1 n# , n œ 1, 2, á 16. an œ ("n)# n1 n 1 , n œ 1, 2, á 17. an œ 3a2n 2b , n œ 1, 2, á 5 18. an œ n2n an 1b , n œ 1, 2, á 19. an œ n# 1, n œ 1, 2, á 20. an œ n 4 , n œ 1, 2, á 21. an œ 4n 3, n œ 1, 2, á 22. an œ 4n 2 , n œ 1, 2, á 23. an œ 3nn! 2 , n œ 1, 2, á 24. an œ 5nn 1 , n œ 1, 2, á 3 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 570 Chapter 10 Infinite Sequences and Series 25. an œ 1 (#1) n 1 , n œ 1, 2, á 26. an œ 27. n lim 2 (0.1)n œ 2 Ê converges Ä_ n "# (1)n ˆ "# ‰ œ Ú n# Û, n œ 1, 2, á # (Theorem 5, #4) n (") 28. n lim œ n lim 1 (n1) œ 1 Ê converges n Ä_ Ä_ n n ˆ"‰ 2 " 2n 2 n 29. n lim œ n lim œ n lim œ 1 Ê converges Ä _ 1 #n Ä _ ˆ"‰ 2 Ä_ # n 2n " 30. n lim œ n lim Ä _ 1 3È n Ä_ % " 5n 31. n lim œ n lim Ä _ n% 8n$ Ä_ 2Èn Š È"n ‹ Š È"n 3‹ Š n"% ‹ 5 1 ˆ 8n ‰ œ _ Ê diverges œ 5 Ê converges n3 n3 " 32. n lim œ n lim œ n lim œ 0 Ê converges Ä _ n# 5n 6 Ä _ (n 3)(n 2) Ä _ n# (n 1)(n 1) n 2n 1 33. n lim œ n lim œ n lim (n 1) œ _ Ê diverges n1 n1 Ä_ Ä_ Ä_ # Š "# ‹ n $ "n n 34 n lim œ n lim œ _ Ê diverges Ä _ 70 4n# Ä _ Š 70# ‹ 4 n 36. n lim (1)n ˆ1 "n ‰ does not exist Ê diverges Ä_ 35. n lim a1 (1)n b does not exist Ê diverges Ä_ ˆ n #n " ‰ ˆ1 "n ‰ œ lim ˆ "# #"n ‰ ˆ1 n" ‰ œ "# Ê converges 37. n lim Ä_ nÄ_ nb1 (") 39. n lim œ 0 Ê converges Ä _ #n 1 ˆ2 #"n ‰ ˆ3 #"n ‰ œ 6 Ê converges 38. n lim Ä_ ) ˆ "# ‰ œ lim (" 40. n lim œ 0 Ê converges Ä_ n Ä _ #n n n 2n É n 2n 41. n lim œ Ên lim Š 2 ‹ œ È2 Ê converges 1 œ É n lim Ä_ Ä _ n1 Ä _ 1 " n " ˆ "0 ‰ œ _ Ê diverges 42. n lim œ n lim Ä _ (0.9)n Ä_ 9 n ˆ 1 n" ‰‹ œ sin 1# œ 1 Ê converges 43. n lim sin ˆ 1# "n ‰ œ sin Šn lim Ä_ Ä_ # 44. n lim n1 cos (n1) œ n lim (n1)(1)n does not exist Ê diverges Ä_ Ä_ sin n 45. n lim œ 0 because n" Ÿ sinn n Ÿ n" Ê converges by the Sandwich Theorem for sequences Ä_ n # # sin n 46. n lim œ 0 because 0 Ÿ sin#n n Ÿ #"n Ê converges by the Sandwich Theorem for sequences Ä _ #n n " ^ 47. n lim œ n lim œ 0 Ê converges (using l'Hopital's rule) Ä _ #n Ä _ #n ln 2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.1 Sequences n n n # $ n 3 (ln 3) 3 (ln 3) 3 3 ln 3 ^ 48. n lim œ n lim œ n lim œ n lim œ _ Ê diverges (using l'Hopital's rule) 6n 6 Ä _ n$ Ä _ 3n# Ä_ Ä_ ln (n ") 49. n lim œ n lim Èn Ä_ Ä_ ˆn " 1‰ " ‹ Š #È n 2È n œ n lim œ n lim Ä _ n1 Ä_ Š È2n ‹ 1 Š n" ‹ œ 0 Ê converges ˆ"‰ ln n n 50. n lim œ n lim œ 1 Ê converges Ä _ ln 2n Ä_ ˆ2‰ 2n 51. n lim 81În œ 1 Ê converges Ä_ (Theorem 5, #3) 52. n lim (0.03)1În œ 1 Ê converges Ä_ (Theorem 5, #3) ˆ1 7n ‰ œ e( Ê converges 53. n lim Ä_ (Theorem 5, #5) n n ) " ˆ1 "n ‰ œ lim ’1 (" 54. n lim Ê converges n “ œe Ä_ nÄ_ n (Theorem 5, #5) n È 55. n lim 10n œ n lim 101În † n1În œ 1 † 1 œ 1 Ê converges Ä_ Ä_ # n n È ˆÈ 56. n lim n# œ n lim n‰ œ 1# œ 1 Ê converges Ä_ Ä_ ˆ3‰ 57. n lim Ä_ n 1 În Ä_ 1În œ " œ 1 Ê converges œ nlim 1 n lim 31În n (Theorem 5, #3 and #2) (Theorem 5, #2) (Theorem 5, #3 and #2) Ä_ 58. n lim (n 4)1ÎÐn4Ñ œ x lim x1Îx œ 1 Ê converges; (let x œ n 4, then use Theorem 5, #2) Ä_ Ä_ ln n Ä_ 1În œ _ œ _ Ê diverges 59. n lim œ nlim 1 n Ä _ n1În lim ln n n Ä_ (Theorem 5, #2) n 60. n lim cln n ln (n 1)d œ n lim ln ˆ n n 1 ‰ œ ln Šn lim ‹ œ ln 1 œ 0 Ê converges Ä_ Ä_ Ä _ n1 n n È 61. n lim 4n n œ n lim 4È n œ 4 † 1 œ 4 Ê converges Ä_ Ä_ (Theorem 5, #2) n È 62. n lim 32n1 œ n lim 32 a1Înb œ n lim 3# † 31În œ 9 † 1 œ 9 Ê converges Ä_ Ä_ Ä_ "†2†3â(n 1)(n) n! ˆ " ‰ œ 0 and nn!n 63. n lim œ n lim Ÿ n lim n†n†nân†n Ä _ nn Ä_ Ä_ n (4) 64. n lim œ 0 Ê converges Ä _ n! n " œ_ 'n Š (10n! ) ‹ n! 66. n lim œ n lim Ä _ 2n 3n Ä_ " œ_ ˆ 6n!n ‰ 1ÎÐln nÑ n! 0 Ê n lim œ 0 Ê converges Ä _ nn (Theorem 5, #6) n! 65. n lim œ n lim Ä _ 106n Ä_ ˆ"‰ 67. n lim Ä_ n (Theorem 5, #3) Ê diverges Ê diverges (Theorem 5, #6) (Theorem 5, #6) œ n lim exp ˆ ln"n ln ˆ n" ‰‰ œ n lim exp ˆ ln 1lnnln n ‰ œ e" Ê converges Ä_ Ä_ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 571 572 Chapter 10 Infinite Sequences and Series n ˆ1 n" ‰n ‹ œ ln e œ 1 Ê converges 68. n lim ln ˆ1 "n ‰ œ ln Šn lim Ä_ Ä_ (Theorem 5, #5) " ‰‰ ˆ 3n " ‰ œ lim exp ˆn ln ˆ 3n 69. n lim œ n lim exp Š ln (3n 1) " ln (3n 1) ‹ 3n 1 Ä _ 3n 1 nÄ_ Ä_ n n 3 3 6n #Î$ ˆ6‰ œ n lim exp 3n 1 "3n 1 œ n lim exp Š (3n 1)(3n Ê converges 1) ‹ œ exp 9 œ e Ä_ Ä_ Š ‹ # n# " ˆ n ‰ œ lim exp ˆn ln ˆ n n 1 ‰‰ œ lim exp Š ln n ln" (n 1) ‹ œ lim exp n 70. n lim ˆ ‰ Ä _ n1 nÄ_ nÄ_ nÄ_ n n"1 Š n"# ‹ n # œ n lim exp Š n(nn 1) ‹ œ e" Ê converges Ä_ n ˆ x ‰ 71. n lim Ä _ 2n 1 1 În œ n lim x ˆ #n " 1 ‰ Ä_ 1 În 1) œ x n lim exp ˆ n" ln ˆ #n " 1 ‰‰ œ x n lim exp Š ln (2n ‹ n Ä_ Ä_ œ x n lim exp ˆ 2n21 ‰ œ xe! œ x, x 0 Ê converges Ä_ n ˆ1 n"# ‰ œ lim exp ˆn ln ˆ1 n"# ‰‰ œ lim exp 72. n lim Ä_ nÄ_ nÄ_ ln Š1 n"# ‹ ˆ n" ‰ Š 2$ ‹‚Š1 n"# ‹ exp – n œ n lim Ä_ Š n"# ‹ — œ n lim exp ˆ n# 2n1 ‰ œ e! œ 1 Ê converges Ä_ 3 †6 36 73. n lim œ n lim œ 0 Ê converges Ä _ 2cn †n! Ä _ n! n n n ˆ 10 ‰n (Theorem 5, #6) ˆ 12 ‰n ˆ 10 ‰n ˆ 120 ‰n 11 11 11 121 74. n lim lim lim œ 0 Ê converges n n œ n n n n œ n Ä _ ˆ 9 ‰ ˆ 11 ‰ n Ä _ ˆ 12 ‰ ˆ 9 ‰ ˆ 12 ‰ ˆ 11 ‰ n Ä _ ˆ 108 ‰ 1 10 12 11 10 11 12 110 (Theorem 5, #4) e e e " 2e 75. n lim tanh n œ n lim œ n lim œ n lim œ n lim " œ 1 Ê converges Ä_ Ä _ en e n Ä _ e2n 1 Ä _ 2e2n Ä_ n n e 76. n lim sinh (ln n) œ n lim Ä_ Ä_ 77. n lim Ä_ 2n ln n 2n n ˆ "n ‰ e ln n œ n lim œ_ 2 # Ä_ Ê diverges ˆcos ˆ n" ‰‰ Š n"# ‹ cos ˆ n" ‰ sin ˆ "n ‰ n# sin ˆ "n ‰ œ lim œ lim œ "# 2n 1 œ n lim Ä _ Š2 " ‹ nÄ_ n Ä _ # ˆ 2n ‰ Š 2# 2$ ‹ # n n 78. n lim n ˆ1 cos "n ‰ œ n lim Ä_ Ä_ Èn sinŠ È1 ‹ œ lim 79. n lim n Ä_ nÄ_ n Ê converges n sin ˆ n" ‰‘ Š "# ‹ ˆ" cos "n ‰ n lim sin ˆ "n ‰ œ 0 Ê converges œ œ n lim ˆ n" ‰ nÄ_ Ä_ Š"‹ n# sinŠ È1n ‹ Èn 1 œ n lim Ä_ cos Š È1n ‹Š 1 2n3Î2 1 ‹ 2n3Î2 œ n lim cos Š È1n ‹ œ cos 0 œ 1 Ê converges Ä_ 80. n lim a3n 5n b1În œ n lim exp’lna3n 5n b1În “ œ n lim exp’ lna3 n 5 b “ œ n lim exp– Ä_ Ä_ Ä_ Ä_ n n œ n lim exp’ Ä_ Š 35n ‹ln 3 ln 5 ˆ 35nn ‰ 1 n 3n ln 3 b 5n ln 5 3n b 5n 1 — ˆ 3 ‰n ln 3 ln 5 exp’ 5 ˆ 3 ‰n 1 “ œ expaln 5b œ 5 “ œ n lim Ä_ 81. n lim tan" n œ 1# Ê converges Ä_ 5 " 82. n lim tan" n œ 0 † 1# œ 0 Ê converges Ä _ Èn Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.1 Sequences 573 n ˆ " ‰n È" n œ lim Šˆ 3" ‰n Š È" ‹ ‹ œ 0 Ê converges 83. n lim Ä_ 3 nÄ_ 2 2 (Theorem 5, #4) # n 1‰ ! È 84. n lim n# n œ n lim exp ’ ln ann nb “ œ n lim exp ˆ 2n n# n œ e œ 1 Ê converges Ä_ Ä_ Ä_ #!! "** (ln n) 85. n lim n Ä_ 200 (ln n) œ n lim n Ä_ & Š 5(lnnn) ‹ 200†199 (ln n) œ n lim n Ä_ "*) 200! œ á œ n lim œ 0 Ê converges Ä_ n % (ln n) 86. n lim œ n lim Ä _ Èn Ä_ – " Š #Èn % $ 10(ln n) 80(ln n) 3840 œ n lim œ n lim œ á œ n lim œ 0 Ê converges Èn Èn Ä_ Ä_ Ä _ Èn ‹ — È # 87. n lim Šn Èn# n‹ œ n lim Šn Èn# n‹ Š n Èn# n ‹ œ n lim Ä_ Ä_ Ä_ n n n n " œ n lim Ä _ 1 É1 " n È n# n n œ "# Ê converges È # È n# 1 È n# n È # " " œ n lim Š ‹ Š Èn# 1 Èn# n ‹ œ n lim 1 n Ä _ È n# 1 È n# n Ä_ n 1 È n# n n 1 n n 88. n lim Ä_ È # œ n lim Ä_ É1 n"# É1 "n ˆ "n 1‰ œ 2 Ê converges " ' " ln n " 89. n lim dx œ n lim œ n lim œ 0 Ê converges Ä_ n 1 x Ä_ n Ä_ n n (Theorem 5, #1) " ˆ " ' " dx œ n lim 90. n lim 1‰ œ p" 1 if p 1 Ê converges ’ " " “ œ n lim Ä _ 1 xp Ä _ 1p xpc1 1 Ä _ 1p npc1 n n 72 2 91. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ê L œ 1 72 L Ê La1 Lb œ 72 Ê L L 72 œ 0 Ä_ n Ä _ n1 Ä _ 1 an Ê L œ 9 or L œ 8; since an 0 for n 1ÊLœ8 an 6 6 2 92. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ê L œ LL 2 Ê LaL 2b œ L 6 Ê L L 6 œ 0 Ä_ n Ä _ n1 Ä _ an 2 Ê L œ 3 or L œ 2; since an 0 for n 2ÊLœ2 È8 2an Ê L œ È8 2L Ê L2 2L 8 œ 0 Ê L œ 2 93. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ or L œ 4; since an 0 for n 3ÊLœ4 È8 2an Ê L œ È8 2L Ê L2 2L 8 œ 0 Ê L œ 2 94. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ or L œ 4; since an 0 for n 2ÊLœ4 È5an Ê L œ È5L Ê L2 5L œ 0 Ê L œ 0 or L œ 5; since 95. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ an 0 for n 1ÊLœ5 ˆ12 Èan ‰ Ê L œ Š12 ÈL‹ Ê L2 25L 144 œ 0 96. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ Ê L œ 9 or L œ 16; since 12 Èan 12 for n 97. an 1 œ 2 a1n , n 1ÊLœ9 1, a1 œ 2. Since an converges Ê n lim a œ L Ê n lim a œ n lim Š2 a1n ‹ Ê L œ 2 L1 Ä_ n Ä _ n1 Ä_ Ê L2 2L 1 œ 0 Ê L œ 1 „ È2; since an 0 for n 1 Ê L œ 1 È2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 574 Chapter 10 Infinite Sequences and Series È 1 an Ê L œ È 1 L 1, a1 œ È1. Since an converges Ê n lim a œ L Ê n lim a œ n lim Ä_ n Ä _ n1 Ä_ 98. an 1 œ È1 an , n È È Ê L2 L 1 œ 0 Ê L œ 1 „2 5 ; since an 0 for n 1 Ê L œ 1 2 5 99. 1, 1, 2, 4, 8, 16, 32, á œ 1, 2! , 2" , 2# , 2$ , 2% , 2& , á Ê x" œ 1 and xn œ 2nc2 for n 2 100. (a) 1# 2(1)# œ 1, 3# 2(2)# œ 1; let f(aß b) œ (a 2b)# 2(a b)# œ a# 4ab 4b# 2a# 4ab 2b# œ 2b# a# ; a# 2b# œ 1 Ê f(aß b) œ 2b# a# œ 1; a# 2b# œ 1 Ê f(aß b) œ 2b# a# œ 1 # # # 2a 4ab 2b " ‰ 2 œ a 4ab 4b(a (b) r#n 2 œ ˆ aa2b œ a(aa b)2b# b œ „" b y# Ê rn œ Ê 2 „ Š yn ‹ b)# # # # # # n n. Let ba represent the (n 1)th fraction where ba In the first and second fractions, yn for n a positive integer lim rn œ È2. 3. Now the nth fraction is aa2b b and a b 2b 2n 2 1 and b n Ê yn n1 n. Thus, nÄ_ 101. (a) f(x) œ x# 2; the sequence converges to 1.414213562 ¸ È2 (b) f(x) œ tan (x) 1; the sequence converges to 0.7853981635 ¸ 14 (c) f(x) œ ex ; the sequence 1, 0, 1, 2, 3, 4, 5, á diverges 102. (a) n lim nf ˆ "n ‰ œ Ä_ f(?x) ?x œ lim ?x Ä !b lim ?x Ä !b f(0?x) f(0) œ f w (0), where ?x œ "n ?x (b) n lim n tan" ˆ n" ‰ œ f w (0) œ 1 " 0# œ 1, f(x) œ tan" x Ä_ (c) n lim n ae1În 1b œ f w (0) œ e! œ 1, f(x) œ ex 1 Ä_ (d) n lim n ln ˆ1 2n ‰ œ f w (0) œ 1 22(0) œ 2, f(x) œ ln (1 2x) Ä_ # # # 103. (a) If a œ 2n 1, then b œ Ú a# Û œ Ú 4n #4n 1 Û œ Ú2n# 2n "# Û œ 2n# 2n, c œ Ü a# Ý œ Ü2n# 2n "# Ý # œ 2n# 2n 1 and a# b# œ (2n 1)# a2n# 2nb œ 4n# 4n 1 4n% 8n$ 4n# # œ 4n% 8n$ 8n# 4n 1 œ a2n# 2n 1b œ c# . (b) a lim Ä_ # Ú a# Û # Ü a# Ý # 2n 2n œ a lim œ 1 or a lim Ä _ 2n# 2n 1 Ä_ # Ú a# Û # Ü a# Ý œ a lim sin ) œ Ä_ lim ) Ä 1 Î2 sin ) œ 1 Š 21 ‹ 2n1 ‰ 104. (a) n lim (2n1)1Î a2nb œ n lim exp ˆ ln2n œ n lim exp 2n#1 œ n lim exp ˆ #"n ‰ œ e! œ 1; Ä_ Ä_ Ä_ Ä_ n n n! ¸ ˆ ne ‰ È 2n1 , Stirlings approximation Ê È n! ¸ ˆ ne ‰ (2n1)1Î a2nb ¸ ne for large values of n (b) n 40 50 60 n È n! 15.76852702 19.48325423 23.19189561 n e 14.71517765 18.39397206 22.07276647 ˆ"‰ ln n " n 105. (a) n lim œ n lim œ n lim œ0 Ä _ nc Ä _ cncc1 Ä _ cnc (b) For all % 0, there exists an N œ eÐln %ÑÎc such that n eÐln %ÑÎc Ê ln n lnc % Ê ln nc ln ˆ "% ‰ Ê nc "% Ê n"c % Ê ¸ n"c 0¸ % Ê lim n"c œ 0 nÄ_ 106. Let {an } and {bn } be sequences both converging to L. Define {cn } by c2n œ bn and c2nc1 œ an , where n œ 1, 2, 3, á . For all % 0 there exists N" such that when n N" then kan Lk % and there exists N# such that when n N# then kbn Lk %. If n 1 2max{N" ß N# }, then kcn Lk %, so {cn } converges to L. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.1 Sequences 575 107. n lim n1În œ n lim exp ˆ "n ln n‰ œ n lim exp ˆ n" ‰ œ e! œ 1 Ä_ Ä_ Ä_ 108. n lim x1În œ n lim exp ˆ "n ln x‰ œ e! œ 1, because x remains fixed while n gets large Ä_ Ä_ 109. Assume the hypotheses of the theorem and let % be a positive number. For all % there exists a N" such that when n N" then kan Lk % Ê % an L % Ê L % an , and there exists a N# such that when n N# then kcn Lk % Ê % cn L % Ê cn L %. If n max{N" ß N# }, then L % an Ÿ bn Ÿ cn L % Ê kbn Lk % Ê n lim b œ L. Ä_ n 110. Let % !. We have f continuous at L Ê there exists $ so that kx Lk $ Ê kf(x) f(L)k %. Also, an Ä L Ê there exists N so that for n N kan Lk $ . Thus for n N, kf(an ) f(L)k % Ê f(an ) Ä f(L). 1) 1 3n 1 3n 4 3n 1 # # an Ê 3(n (n 1) 1 n 1 Ê n # n 1 Ê 3n 3n 4n 4 3n 6n n 2 111. an1 " Ê 4 2; the steps are reversible so the sequence is nondecreasing; 3n n 1 3 Ê 3n 1 3n 3 Ê 1 3; the steps are reversible so the sequence is bounded above by 3 1) 3)! (2n 3)! (2n 5)! (2n 3)! (2n 5)! (n 2)! an Ê (2(n ((n 1) 1)! (n 1)! Ê (n 2)! (n 1)! Ê (2n 3)! (n 1)! 112. an1 Ê (2n 5)(2n 4) n 2; the steps are reversible so the sequence is nondecreasing; the sequence is not 3)! bounded since (2n (n 1)! œ (2n 3)(2n 2)â(n 2) can become as large as we please nb1 nb1 n n nb1 nb1 113. an1 Ÿ an Ê 2(n 31)! Ÿ 2n!3 Ê 2 2n 33n Ÿ (n n!1)! Ê 2 † 3 Ÿ n 1 which is true for n 5; the steps are reversible so the sequence is decreasing after a& , but it is not nondecreasing for all its terms; a" œ 6, a# œ 18, a$ œ 36, a% œ 54, a& œ 324 5 œ 64.8 Ê the sequence is bounded from above by 64.8 an Ê 2 n2 1 #n"b1 114. an1 2 2n #"n Ê 2n n2 1 " " #nb1 #n Ê n(n 2 1) #n"b1 ; the steps are reversible so the sequence is nondecreasing; 2 2n #"n Ÿ 2 Ê the sequence is bounded from above 115. an œ 1 "n converges because n" Ä 0 by Example 1; also it is a nondecreasing sequence bounded above by 1 116. an œ n "n diverges because n Ä _ and n" Ä 0 by Example 1, so the sequence is unbounded 117. an œ 2 2n 1 œ 1 #"n and 0 #"n "n ; since n" Ä 0 (by Example 1) Ê #"n Ä 0, the sequence converges; also it is n a nondecreasing sequence bounded above by 1 n 118. an œ 2 3n 1 œ ˆ 23 ‰ 3"n ; the sequence converges to ! by Theorem 5, #4 n 119. an œ a(1)n 1b ˆ nn 1 ‰ diverges because an œ 0 for n odd, while for n even an œ 2 ˆ1 n" ‰ converges to 2; it diverges by definition of divergence 120. xn œ max {cos 1ß cos 2ß cos 3ß á ß cos n} and xn1 œ max {cos 1ß cos 2ß cos 3ß á ß cos (n 1)} so the sequence is nondecreasing and bounded above by 1 Ê the sequence converges. 121. an È an1 Í 1 Èn 2n È and 1 Èn 2n " È2(n 1) Èn 1 Í Èn 1 È2n# 2n xn with xn Ÿ 1 Èn È2n# 2n Í Èn 1 È2 ; thus the sequence is nonincreasing and bounded below by È2 Ê it converges Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Èn 576 Chapter 10 Infinite Sequences and Series 122. an an1 Í n n 1 (n 1) " n1 Í n# 2n 1 n# 2n Í 1 0 and n n 1 1; thus the sequence is nonincreasing and bounded below by 1 Ê it converges 123. n n n" n 4nb1 3n ˆ 43 ‰n1 Í 1 43 and œ 4 ˆ 43 ‰ so an an1 Í 4 ˆ 43 ‰ 4 ˆ 43 ‰ Í ˆ 43 ‰ 4n n 4 ˆ 34 ‰ 4; thus the sequence is nonincreasing and bounded below by 4 Ê it converges 124. a" œ 1, a# œ 2 3, a$ œ 2(2 3) 3 œ 2# a22 "b † 3, a% œ 2 a2# a22 "b † 3b 3 œ 2$ a2$ 1b 3, a& œ 2 c2$ a2$ 1b 3d 3 œ 2% a2% 1b 3, á , an œ 2n" a2n" 1b 3 œ 2n" 3 † 2n1 3 œ 2n1 (1 3) 3 œ 2n 3; an an1 Í 2n 3 2n1 3 Í 2n 2n1 Í 1 Ÿ 2 so the sequence is nonincreasing but not bounded below and therefore diverges 125. Let 0 M 1 and let N be an integer greater than 1 MM . Then n N Ê n 1 MM Ê n nM M Ê n M nM Ê n M(n 1) Ê n n 1 M. 126. Since M" is a least upper bound and M# is an upper bound, M" Ÿ M# . Since M# is a least upper bound and M" is an upper bound, M# Ÿ M" . We conclude that M" œ M# so the least upper bound is unique. 127. The sequence an œ 1 ("#) is the sequence "# , 3# , "# , 3# , á . This sequence is bounded above by 3# , n but it clearly does not converge, by definition of convergence. 128. Let L be the limit of the convergent sequence {an }. Then by definition of convergence, for #% there corresponds an N such that for all m and n, m N Ê kam Lk #% and n N Ê kan Lk #% . Now kam an k œ kam L L an k Ÿ kam Lk kL an k #% #% œ % whenever m N and n N. 129. Given an % 0, by definition of convergence there corresponds an N such that for all n N, kL" an k % and kL# an k %. Now kL# L" k œ kL# an an L" k Ÿ kL# an k kan L" k % % œ 2%. kL# L" k 2% says that the difference between two fixed values is smaller than any positive number 2%. The only nonnegative number smaller than every positive number is 0, so kL" L# k œ 0 or L" œ L# . 130. Let k(n) and i(n) be two order-preserving functions whose domains are the set of positive integers and whose ranges are a subset of the positive integers. Consider the two subsequences akÐnÑ and aiÐnÑ , where akÐnÑ Ä L" , aiÐnÑ Ä L# and L" Á L# . Thus ¸akÐnÑ aiÐnÑ ¸ Ä kL" L# k 0. So there does not exist N such that for all m, n N Ê kam an k %. So by Exercise 128, the sequence Öan × is not convergent and hence diverges. 131. a2k Ä L Í given an % 0 there corresponds an N" such that c2k N" Ê ka2k Lk %d . Similarly, a2k1 Ä L Í c2k 1 N# Ê ka2k1 Lk %d . Let N œ max{N" ß N# }. Then n N Ê kan Lk % whether n is even or odd, and hence an Ä L. 132. Assume an Ä 0. This implies that given an % 0 there corresponds an N such that n N Ê kan 0k % Ê kan k % Ê kkan kk % Ê kkan k 0k % Ê kan k Ä 0. On the other hand, assume kan k Ä 0. This implies that given an % 0 there corresponds an N such that for n N, kkan k 0k % Ê kkan kk % Ê kan k % Ê kan 0k % Ê an Ä 0. # # # # 133. (a) f(x) œ x# a Ê f w (x) œ 2x Ê xn1 œ xn xn#xn a Ê xn1 œ 2xn 2xaxnn ab œ xn2xn a œ ˆxn xa ‰ # n (b) x" œ 2, x# œ 1.75, x$ œ 1.732142857, x% œ 1.73205081, x& œ 1.732050808; we are finding the positive number where x# 3 œ 0; that is, where x# œ 3, x 0, or where x œ È3 . Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.2 Infinite Series 577 134. x" œ 1, x# œ 1 cos (1) œ 1.540302306, x$ œ 1.540302306 cos (1 cos (1)) œ 1.570791601, x% œ 1.570791601 cos (1.570791601) œ 1.570796327 œ 1# to 9 decimal places. After a few steps, the arc axnc1 b and line segment cos axnc1 b are nearly the same as the quarter circle. 135-146. Example CAS Commands: Mathematica: (sequence functions may vary): Clear[a, n] a[n_]; = n1 / n first25= Table[N[a[n]],{n, 1, 25}] Limit[a[n], n Ä 8] Mathematica: (sequence functions may vary): Clear[a, n] a[n_]; = n1 / n first25= Table[N[a[n]],{n, 1, 25}] Limit[a[n], n Ä 8] The last command (Limit) will not always work in Mathematica. You could also explore the limit by enlarging your table to more than the first 25 values. If you know the limit (1 in the above example), to determine how far to go to have all further terms within 0.01 of the limit, do the following. Clear[minN, lim] lim= 1 Do[{diff=Abs[a[n] lim], If[diff < .01, {minN= n, Abort[]}]}, {n, 2, 1000}] minN For sequences that are given recursively, the following code is suggested. The portion of the command a[n_]:=a[n] stores the elements of the sequence and helps to streamline computation. Clear[a, n] a[1]= 1; a[n_]; = a[n]= a[n 1] (1/5)(n1) first25= Table[N[a[n]], {n, 1, 25}] The limit command does not work in this case, but the limit can be observed as 1.25. Clear[minN, lim] lim= 1.25 Do[{diff=Abs[a[n] lim], If[diff < .01, {minN= n, Abort[]}]}, {n, 2, 1000}] minN 10.2 INFINITE SERIES n a1 r b 1. sn œ a (1 r) œ 2 ˆ1 ˆ "3 ‰ ‰ 1 ˆ "3 ‰ a1 r b 2. sn œ a (1 r) œ 9 ‰ˆ " ‰n ‰ ˆ 100 1 ˆ 100 " 1 ˆ 100 ‰ n n 1 ˆ " ‰ Ê n lim s œ 1 2ˆ " ‰ œ 3 Ä_ n 3 ˆ 9 ‰ " Ê n lim s œ 1 100 ˆ " ‰ œ 11 Ä_ n 100 n a1 r b # 3. sn œ a (1 s œ ˆ "3 ‰ œ 32 r) œ 1 ˆ "# ‰ Ê n lim Ä_ n # n 4. sn œ 11 ((2)2) , a geometric series where krk 1 Ê divergence n 5. " " " (n 1)(n #) œ n 1 n # Ê sn œ ˆ #" 3" ‰ ˆ 3" 4" ‰ á ˆ n " 1 n " # ‰ œ #" n " # Ê n lim s œ #" Ä_ n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 578 6. Chapter 10 Infinite Sequences and Series 5 5 5 n(n 1) œ n n 1 Ê sn œ ˆ5 52 ‰ ˆ 52 53 ‰ ˆ 53 54 ‰ á ˆ n 5 1 n5 ‰ ˆ n5 n 5 1 ‰ œ 5 n 5 1 Ê n lim s œ5 Ä_ n " " 7. 1 "4 16 64 á , the sum of this geometric series is 1 ˆ" " ‰ œ 1 "ˆ " ‰ œ 45 4 4 8. " ‰ ˆ 16 " " " " 16 64 256 á , the sum of this geometric series is 1 ˆ 4" ‰ œ 1# 9. ˆ 74 ‰ 7 7 7 7 4 16 64 á , the sum of this geometric series is 1 ˆ "4 ‰ œ 3 5 5 10. 5 54 16 64 á , the sum of this geometric series is 1 ˆ5 " ‰ œ 4 4 11. (5 1) ˆ 5# "3 ‰ ˆ 45 9" ‰ ˆ 85 #"7 ‰ á , is the sum of two geometric series; the sum is 5 1 "ˆ " ‰ œ 10 #3 œ 23 # 1 ˆ "# ‰ 3 12. (5 1) ˆ 5# "3 ‰ ˆ 45 9" ‰ ˆ 85 #"7 ‰ á , is the difference of two geometric series; the sum is 5 1 "ˆ " ‰ œ 10 #3 œ 17 # 1 ˆ "# ‰ 3 " ‰ 13. (1 1) ˆ 1# "5 ‰ ˆ 41 25 ˆ 18 1#"5 ‰ á , is the sum of two geometric series; the sum is 1 1 "ˆ " ‰ œ 2 56 œ 17 6 1 ˆ "# ‰ 5 8 16 4 8 14. 2 45 25 125 á œ 2 ˆ1 25 25 125 á ‰ ; the sum of this geometric series is 2 Š 1 "ˆ 2 ‰ ‹ œ 10 3 5 15. Series is geometric with r œ 25 Ê ¹ 25 ¹ 1 Ê Converges to 1 1 2 œ 53 5 16. Series is geometric with r œ 3 Ê ¹3¹ 1 Ê Diverges 1 17. Series is geometric with r œ 18 Ê ¹ 18 ¹ 1 Ê Converges to 1 8 1 œ 17 8 2 18. Series is geometric with r œ 23 Ê ¹ 23 ¹ 1 Ê Converges to 1 ˆ3 2 ‰ œ 25 3 _ n _ Š 23 ‹ 23 ˆ " ‰ 23 19. 0.23 œ ! 100 œ 1 100 œ 99 10# ˆ " ‰ nœ0 nœ0 100 _ n 7 ˆ " ‰ 21. 0.7 œ ! 10 œ 10 nœ0 _ 7 Š 10 ‹ " 1 Š 10 ‹ n 1 ‰ˆ 6 ‰ˆ " ‰ 23. 0.06 œ ! ˆ 10 œ 10 10 nœ0 _ n nœ0 6 Š 100 ‹ " 1 Š 10 ‹ 414 ˆ " ‰ 24. 1.414 œ 1 ! 1000 œ1 10$ nœ0 _ n d ˆ " ‰ 22. 0.d œ ! 10 œ 10 œ 79 234 Š 1000 ‹ n 234 ˆ " ‰ 20. 0.234 œ ! 1000 œ 10$ " 1 Š 1000 ‹ d Š 10 ‹ " 1 Š 10 ‹ œ d9 6 " œ 90 œ 15 414 Š 1000 ‹ " 1 Š 1000 ‹ "413 œ 1 414 999 œ 999 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 234 œ 999 Section 10.2 Infinite Series _ Š 123& ‹ n ! 123& ˆ " $ ‰ œ 124 25. 1.24123 œ 124 100 10 10 100 10 1 Š "$ ‹ nœ0 _ Š 142,857 ' ‹ n 10 1 Š "' ‹ nœ0 28. lim n œ lim 11 œ 1 Á 0 Ê diverges nÄ_ n 10 nÄ_ lim n an 1 b 2n 1 2 n œ lim n2 n 5n 6 œ lim 2n 5 œ lim 2 œ 1 Á 0 Ê diverges 2 nÄ_ an 2ban 3b nÄ_ nÄ_ lim 1 œ 0 Ê test inconclusive nÄ_ n 4 30. 1 lim 2 n œ lim 2n œ 0 Ê test inconclusive nÄ_ n 3 nÄ_ 32. 33. 34. 3,142,854 116,402 œ 3 142,857 10' 1 œ 999,999 œ 37,037 10 29. 31. 123,999 41,333 124 124 123 œ 100 10&123 10# œ 100 99,900 œ 99,900 œ 33,300 10 ˆ 10" ' ‰ œ 3 26. 3.142857 œ 3 ! 142,857 10' 27. 579 nÄ_ lim cos 1n œ cos 0 œ 1 Á 0 Ê diverges nÄ_ n lim ne œ nÄ_ e n n lim n e nÄ_ e 1 n œ lim een œ lim 11 œ 1 Á 0 Ê diverges nÄ_ nÄ_ lim ln 1n œ _ Á 0 Ê diverges nÄ_ lim cos n 1 œ does not exist Ê diverges nÄ_ 35. sk œ ˆ1 2" ‰ ˆ 2" 3" ‰ ˆ 3" 4" ‰ á ˆ k " 1 k" ‰ ˆ k" k " 1 ‰ œ 1 k " 1 Ê œ lim ˆ1 k " 1 ‰ œ 1, series converges to 1 lim sk kÄ_ kÄ_ 3 ‰ 36. sk œ ˆ 31 34 ‰ ˆ 34 39 ‰ ˆ 39 16 á Š ak 3 1b2 k32 ‹ Š k32 ak 3 1b2 ‹ œ 3 ak 3 1b2 Ê lim sk kÄ_ œ lim Š3 ak 3 1b2 ‹ œ 3, series converges to 3 kÄ_ 37. sk œ ŠlnÈ2 lnÈ1‹ ŠlnÈ3 lnÈ2‹ ŠlnÈ4 lnÈ3‹ á ŠlnÈk lnÈk 1‹ ŠlnÈk 1 lnÈk‹ œ lnÈk 1 lnÈ1 œ lnÈk 1 Ê lim sk œ lim lnÈk 1 œ _; series diverges kÄ_ kÄ_ 38. sk œ atan 1 tan 0b atan 2 tan 1b atan 3 tan 2b á atan k tan ak 1bb atan ak 1b tan kb œ tan ak 1b tan 0 œ tan ak 1b Ê lim sk œ lim tan ak 1b œ does not exist; series diverges kÄ_ kÄ_ 39. sk œ ˆcos1 ˆ 12 ‰ cos1 ˆ 13 ‰‰ ˆcos1 ˆ 13 ‰ cos1 ˆ 14 ‰‰ ˆcos1 ˆ 14 ‰ cos1 ˆ 15 ‰‰ á ˆcos1 ˆ 1k ‰ cos1 ˆ k 1 1 ‰‰ ˆcos1 ˆ k 1 1 ‰ cos1 ˆ k 1 # ‰‰ œ 13 cos1 ˆ k 1 # ‰ Ê lim sk œ lim ’ 13 cos1 ˆ k 1 # ‰“ œ 13 12 œ 16 , series converges to 16 kÄ_ kÄ_ 40. sk œ ŠÈ5 È4‹ ŠÈ6 È5‹ ŠÈ7 È6‹ á ŠÈk 3 Èk 2‹ ŠÈk 4 Èk 3‹ œ Èk 4 2 Ê lim sk œ lim ’Èk 4 2“ œ _; series diverges kÄ_ kÄ_ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 580 Chapter 10 Infinite Sequences and Series 4 " " "‰ " ‰ " ‰ ˆ ˆ" "‰ ˆ" ˆ " 41. (4n 3)(4n 1) œ 4n 3 4n 1 Ê sk œ 1 5 5 9 9 13 á 4k 7 4k 3 ˆ 4k " 3 4k " 1 ‰ œ 1 4k " 1 Ê lim sk œ lim ˆ1 4k " 1 ‰ œ 1 kÄ_ kÄ_ A(2n 1) B(2n 1) 6 A B 42. (2n 1)(2n Ê A(2n 1) B(2n 1) œ 6 Ê (2A 2B)n (A B) œ 6 1) œ 2n 1 2n 1 œ (2n 1)(2n 1) Ê œ k k 2A 2B œ 0 ABœ0 6 Êœ Ê 2A œ 6 Ê A œ 3 and B œ 3. Hence, ! (2n 1)(2n œ 3 ! ˆ #n " 1 #n " 1 ‰ 1) A Bœ6 ABœ6 nœ1 nœ1 œ 3 Š "1 3" 3" 5" 5" 7" á #(k "1) 1 2k " 1 #k " 1 ‹ œ 3 ˆ1 #k " 1 ‰ Ê the sum is lim 3 ˆ1 #k " 1 ‰ œ 3 kÄ_ # # # # A(2n1)(2n1) B(2n1) C(2n1)(2n1) D(2n1) A B C D 43. (2n1)40n # (2n1)# œ (2n1) (2n1)# (2n1) (2n1)# œ (2n1)# (2n1)# Ê A(2n 1)(2n 1)# B(2n 1)# C(2n 1)(2n 1)# D(2n 1)# œ 40n Ê A a8n$ 4n# 2n 1b B a4n# 4n 1b C a8n$ 4n# 2n 1b œ D a4n# 4n 1b œ 40n Ê (8A 8C)n$ (4A 4B 4C 4D)n# (2A 4B 2C 4D)n (A B C D) œ 40n Ú Ú 8A 8C œ 0 8A 8C œ 0 Ý Ý Ý Ý 4A 4B 4C 4D œ 0 A BC Dœ 0 B Dœ 0 Ê Û Ê Û Ê œ Ê 4B œ 20 Ê B œ 5 œ 2A 4B 2C 4D 40 A 2 œ 2D œ 20 B C 2D 20 2B Ý Ý Ý Ý Ü A B C D œ 0 Ü A B C D œ 0 and D œ 5 Ê œ k ACœ0 Ê C œ 0 and A œ 0. Hence, ! ’ (#n1)40n # (2n1)# “ A 5 C 5 œ 0 nœ1 k œ 5 ! ’ (#n" 1)# (#n" 1)# “ œ 5 Š 1" 9" 9" #"5 #"5 á (2(k1)" 1)# (#k" 1)# (#k" 1)# ‹ nœ1 œ 5 Š1 (2k" 1)# ‹ Ê the sum is n lim 5 Š1 (2k" 1)# ‹ œ 5 Ä_ 1 " " "‰ " ‰ " " " " ˆ ˆ" "‰ ˆ" 44. n#2n (n 1)# œ n# (n 1)# Ê sk œ 1 4 4 9 9 16 á ’ (k 1)# k# “ ’ k# (k 1)# “ Ê lim sk œ lim ’1 (k " 1)# “ œ 1 kÄ_ kÄ_ 45. sk œ Š1 È" ‹ Š È" È"3 ‹ Š È"3 È" ‹ á Š È " 2 Ê 2 4 k1 È" ‹ Š È" Èk" 1 ‹ œ 1 Èk" 1 k k lim sk œ lim Š1 Èk" 1 ‹ œ 1 kÄ_ kÄ_ " " ‰ " " ‰ 46. sk œ ˆ "# #""Î# ‰ ˆ #"Î# #"Î$ ˆ #"Î$ #"Î% á ˆ #1ÎÐ"k 1Ñ #1"Îk ‰ ˆ #1"Îk #1ÎÐ"k1Ñ ‰ œ #" #1ÎÐ"k1Ñ Ê lim sk œ "# 1" œ #" kÄ_ 47. sk œ ˆ ln"3 ln"# ‰ ˆ ln"4 ln"3 ‰ ˆ ln"5 ln"4 ‰ á Š ln (k" 1) ln"k ‹ Š ln (k" 2) ln (k" 1) ‹ œ ln"# ln (k" 2) Ê lim sk œ ln"# kÄ_ 48. sk œ ctan" (1) tan" (2)d ctan" (2) tan" (3)d á ctan" (k 1) tan" (k)d ctan" (k) tan" (k 1)d œ tan" (1) tan" (k 1) Ê lim sk œ tan" (1) 1# œ 14 1# œ 14 kÄ_ 49. convergent geometric series with sum È " œ È22 1 œ 2 È2 " 1 ŠÈ ‹ 2 50. divergent geometric series with krk œ È2 1 51. convergent geometric series with sum Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Š 3# ‹ 1 Š "# ‹ œ1 Section 10.2 Infinite Series 52. n lim (1)n1 n Á 0 Ê diverges Ä_ 581 53. n lim cos (n1) œ n lim (1)n Á 0 Ê diverges Ä_ Ä_ 54. cos (n1) œ (1)n Ê convergent geometric series with sum 55. convergent geometric series with sum " œ 56 1 Š "5 ‹ # " œ e#e1 1 Š "# ‹ e 56. n lim ln 3"n œ _ Á 0 Ê diverges Ä_ 57. convergent geometric series with sum 2 18 2 2 œ 20 " 9 9 œ 9 1 Š 10 ‹ 58. convergent geometric series with sum " œ x x 1 1 Š "x ‹ 59. difference of two geometric series with sum n " " " œ 3 3# œ 3# 1 Š 23 ‹ 1 Š3‹ n ˆ1 "n ‰ œ lim ˆ1 " ‰ œ e" Á 0 Ê diverges 60. n lim n Ä_ nÄ_ n n†nân 62. n lim œ n lim n lim n œ _ Ê diverges Ä _ n! Ä _ 1†#ân Ä_ n n! 61. n lim œ _ Á 0 Ê diverges Ä _ 1000n _ _ _ _ _ n n _ n _ n 63. ! 2 4n 3 œ ! 42n ! 43n œ ! ˆ 21 ‰ ! ˆ 43 ‰ ; both œ ! ˆ 21 ‰ and ! ˆ 43 ‰ are geometric series, and both converge n n nœ1 n nœ1 n nœ1 nœ1 nœ1 nœ1 nœ1 _ 1 _ 3 n n since r œ 12 Ê ¹ 12 ¹ 1 and r œ 34 Ê ¹ 34 ¹ 1, respectivley Ê ! ˆ 12 ‰ œ 1 2 1 œ 1 and ! ˆ 34 ‰ œ 1 4 3 œ 3 Ê 2 4 nœ1 _ nœ1 ! 2 n 3 œ 1 3 œ 4 by Theorem 8, part (1) nœ1 64. n n 4 4 lim 23n 4n œ n 2n n ˆ 1 ‰n " " lim 43nn " œ lim ˆ 23 ‰n " œ 11 œ 1 Á 0 Ê diverges by nth term test for divergence nÄ_ nÄ_ 4n _ _ nœ1 nœ1 nÄ_ 4 65. ! ln ˆ n n 1 ‰ œ ! cln (n) ln (n 1)d Ê sk œ cln (1) ln (2)d cln (2) ln (3)d cln (3) ln (4)d á cln (k 1) ln (k)d cln (k) ln (k 1)d œ ln (k 1) Ê lim sk œ _, Ê diverges kÄ_ 66. n lim a œ n lim ln ˆ 2n n 1 ‰ œ ln ˆ #" ‰ Á 0 Ê diverges Ä_ n Ä_ 1 67. convergent geometric series with sum 1 "ˆ e ‰ œ 1 e 1 1 68. divergent geometric series with krk œ 1e e ¸ 23.141 22.459 1 _ _ nœ0 nœ0 _ _ nœ0 nœ0 69. ! (1)n xn œ ! (x)n ; a œ 1, r œ x; converges to 1 "(x) œ 1 " x for kxk 1 70. ! (1)n x2n œ ! ax# b ; a œ 1, r œ x# ; converges to 1 " x# for kxk 1 n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 582 Chapter 10 Infinite Sequences and Series 71. a œ 3, r œ x # 1 ; converges to _ _ 72. ! (#1) ˆ 3 "sin x ‰ œ ! n n nœ0 nœ0 3 œ 3 6 x for 1 x" # 1 or 1 x 3 1 Šx # "‹ ˆ "# ‰ " ˆ " ‰n " " " # 3 sin x ; a œ # , r œ 3 sin x ; converges to 1 Š 3 sin x ‹ 3 sin x 3 sin x " " " ˆ ‰ œ 2(4 sin x) œ 8 2 sin x for all x since 4 Ÿ 3 sin x Ÿ # for all x 73. a œ 1, r œ 2x; converges to 1 " 2x for k2xk 1 or kxk #" 74. a œ 1, r œ x"# ; converges to # " œ x#x 1 for ¸ x1# ¸ 1 or kxk 1. 1 Š #" ‹ x 75. a œ 1, r œ (x 1)n ; converges to 1 (x" 1) œ # " x for kx 1k 1 or 2 x 0 76. a œ 1, r œ 3 # x ; converges to " œ x 2 1 for ¸ 3 # x ¸ 1 or 1 x 5 1 Š3 # x‹ 77. a œ 1, r œ sin x; converges to 1 "sin x for x Á (2k 1) 1# , k an integer 78. a œ 1, r œ ln x; converges to 1 "ln x for kln xk 1 or e" x e _ 79. (a) ! nœ2 _ 80. (a) ! nœ1 _ " (n 4)(n 5) " (b) ! (n 2)(n 3) 5 (n 2)(n 3) 5 (b) ! (n 2)(n 1) nœ0 Š "# ‹ 1 Š "# ‹ n 1 Š 3# ‹ 1 Š "# ‹ nœ0 _ _ nœ1 nœ1 nœ1 _ _ _ nœ1 nœ1 nœ1 5 (n 19)(n 18) œ 3 Š "# ‹ 1 Š "# ‹ is a geometric series whose sum is _ nœ20 œ1 " (c) one example is 1 "# 4" 8" 16 á œ1 n _ (c) ! nœ3 3 (b) one example is 3# 43 83 16 á œ _ nœ5 _ " 81. (a) one example is "# 4" 8" 16 á œ 82. The series ! kˆ 12 ‰ _ " (c) ! (n 3)(n #) œ 0. Š k# ‹ 1 Š "# ‹ n œ k where k can be any positive or negative number. _ _ nœ1 nœ1 _ _ nœ1 nœ1 83. Let an œ bn œ ˆ "# ‰ . Then ! an œ ! bn œ ! ˆ "# ‰ œ 1, while ! Š bann ‹ œ ! (1) diverges. n n n 84. Let an œ bn œ ˆ "# ‰ . Then ! an œ ! bn œ ! ˆ "# ‰ œ 1, while ! aan bn b œ ! ˆ 4" ‰ œ 3" Á AB. n n _ _ _ _ nœ1 nœ1 nœ1 nœ1 n 85. Let an œ ˆ "4 ‰ and bn œ ˆ #" ‰ . Then A œ ! an œ 3" , B œ ! bn œ 1 and ! Š bann ‹ œ ! ˆ #" ‰ œ 1 Á AB . 86. Yes: ! Š a"n ‹ diverges. The reasoning: ! an converges Ê an Ä 0 Ê a"n Ä _ Ê ! Š a"n ‹ diverges by the nth-Term Test. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.3 The Integral Test 87. Since the sum of a finite number of terms is finite, adding or subtracting a finite number of terms from a series that diverges does not change the divergence of the series. 88. Let An œ a" a# á an and n lim A œ A. Assume ! aan bn b converges to S. Let Ä_ n Sn œ (a" b" ) (a# b# ) á (an bn ) Ê Sn œ (a" a# á an ) (b" b# á bn ) Ê b" b# á bn œ Sn An Ê n lim ab" b# á bn b œ S A Ê ! bn converges. This Ä_ contradicts the assumption that ! bn diverges; therefore, ! aan bn b diverges. 89. (a) (b) 2 1r œ 5 Š 13 2 ‹ 1r # Ê 52 œ 1 r Ê r œ 53 ; 2 2 ˆ 53 ‰ 2 ˆ 53 ‰ á # $ 13 3 13 ˆ 3 ‰ 13 ˆ 3 ‰ ˆ 3 ‰ á œ 5 Ê 10 œ 1 r Ê r œ 10 ; 13 13 2 # 10 # 10 # 10 90. 1 eb e2b á œ 1"eb œ 9 Ê 9" œ 1 eb Ê eb œ 98 Ê b œ ln ˆ 98 ‰ 91. sn œ 1 2r r# 2r$ r% 2r& á r2n 2r2n1 , n œ 0, 1, á Ê sn œ a1 r# r% á r2n b a2r 2r$ 2r& á 2r2n1 b Ê n lim s œ 1 " r# 1 2rr# Ä_ n 2r # œ 11 r# , if kr k 1 or krk 1 92. L sn œ 1 a r a a11 rr b œ 1arr n n # # 93. area œ 2# ŠÈ2‹ (1)# Š È" ‹ á œ 4 2 1 #" á œ 1 4 " œ 8 m# 2 # # 94. (a) L" œ 3, L# œ 3 ˆ 43 ‰ , L$ œ 3 ˆ 43 ‰ , á , Ln œ 3 ˆ 43 ‰ (b) nc1 Ê n lim L œ n lim 3 ˆ 43 ‰ Ä_ n Ä_ nc1 œ_ È È Using the fact that the area of an equilateral triangle of side length s is 43 s2 , we see that A" œ 43 , È È È È È È È 2 2 A# œ A" 3Š 43 ‹ˆ "3 ‰ œ 43 1#3 , A$ œ A# 3a4bŠ 43 ‹ˆ 3"2 ‰ œ 43 123 #73 , È È 2 2 A% œ A$ 3a4b2 Š 43 ‹ˆ 3"3 ‰ , A5 œ A4 3a4b3 Š 43 ‹ˆ 3"4 ‰ , . . . , An œ n n n È3 ! 3a4bk2 Š È3 ‹ˆ "2 ‰k1 œ È3 ! 3È3a4bk$ ˆ " ‰k1 œ È3 3È3Œ! 4kkc$1 . 4 4 3 4 9 4 9 kœ2 kœ2 kœ2 n 1 È3 kc$ È È È 1 ‰ lim An œ n lim 3È3Œ! 49k 1 œ 43 3È3Œ 1 36 4 œ 43 3È3ˆ 20 œ 43 ˆ1 53 ‰ nÄ_ Ä_ Œ 4 9 kœ2 È œ 43 ˆ 85 ‰ œ 85 A" 10.3 THE INTEGRAL TEST _1 1; '1 1. faxb œ x12 is positive, continuous, and decreasing for x œ lim bÄ_ _1 ˆ 1b 1‰ œ 1 Ê ' 1 x _ bÄ_ ˆ 54 b0.8 54 ‰ œ _ Ê ' '1b x1 dx œ lim 2 bÄ_ b ’ 1x “ 1 ! 12 converges 2 dx converges Ê n œ1 n _ 1 1; '1 2. faxb œ x10.2 is positive, continuous, and decreasing for x œ lim x2 dx œ b lim Ä_ _ 1 1 x0.2 dx diverges Ê _ x0.2 dx œ b lim Ä_ '1b x1 dx œ lim 0.2 bÄ_ ! 10.2 diverges n œ1 n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. b ’ 45 x0.8 “ 1 583 584 Chapter 10 Infinite Sequences and Series _ 1; '1 3. faxb œ x2 1 4 is positive, continuous, and decreasing for x œ lim ˆ 12 tan1 b2 12 tan1 12 ‰ œ 14 12 tan1 12 Ê bÄ_ '1b x 1 4 dx œ lim ’lnlx 4l“ '1b e2x dx œ lim ’ 12 e2x “ _ 2 _ 1 1; '1 x 4 dx œ b lim Ä_ _ b bÄ_ 1 2 n œ1 _ alnlb 4l ln 5b œ _ Ê '1 x 1 4 dx diverges Ê ! n 1 4 diverges bÄ_ ’ 21 tan1 2x “ 2 '1_ x 1 4 dx converges Ê ! n 1 4 converges 4. faxb œ x 1 4 is positive, continuous, and decreasing for x œ lim '1b x 1 4 dx œ lim 1 x2 4 dx œ b lim Ä_ b bÄ_ 1 n œ1 _ 1; '1 e2x dx œ lim 5. faxb œ e2x is positive, continuous, and decreasing for x _ 2x ˆ 2e12b 2e12 ‰ œ 2e12 Ê ' e 1 œ lim bÄ_ _ bÄ_ _ 2; '2 '3 x x 4 dx œ lim '3 x x 4 dx œ lim 2 bÄ_ _ _ n œ3 n œ1 '2b xaln1xb dx œ lim 2 b ’ ln1x “ bÄ_ 1, f w axb œ ax42x4b2 0 for x 2, thus f is decreasing for x 2 b b 2 1 dx œ lim xaln xb2 bÄ_ 2 1 dx converges Ê ! naln1nb2 converges xaln xb2 n œ2 7. faxb œ x2 x 4 is positive and continuous for x _ 1 n œ1 _ ˆ ln1b ln12 ‰ œ ln12 Ê ' 2 œ lim b bÄ_ dx converges Ê ! e2n converges 6. faxb œ xaln1xb2 is positive, continuous, and decreasing for x _ bÄ_ bÄ_ ’ 21 lnax2 4b“ œ lim bÄ_ 3 _ 3; ˆ 21 lnab2 4b 21 lna13b‰ œ _ Ê ' _ 3 x x2 4 dx diverges Ê ! n2 n 4 diverges Ê ! n2 n 4 œ 51 82 ! n2 n 4 diverges n œ3 2, f w axb œ 2 xln2 x 0 for x e, thus f is decreasing for x 2 2 8. faxb œ lnxx is positive and continuous for x _ ln x2 '3 dx œ lim x bÄ_ _ '3 lnxx dx œ lim b _ 2 _ ln x2 b 2 bÄ_ ’2aln xb“ œ lim _ 2 bÄ_ 3 a2aln bb 2aln 3bb œ _ Ê '3 x 3; dx 2 diverges Ê ! lnnn diverges Ê ! lnnn œ ln24 ! lnnn diverges n œ3 n œ2 n œ3 2 9. faxb œ exxÎ3 is positive and continuous for x _ x2 '7 e xÎ3 dx œ lim bÄ_ 2 xÎ3 bÄ_ 18b lim Š 3a6b ‹ e327 7Î3 œ ebÎ3 œ lim bÄ_ _ '7 ex dx œ lim b bÄ_ ax 6 b 1, f w axb œ x3e 0 for x 6, thus f is decreasing for x xÎ3 b 2 54 lim ’ e3xxÎ3 e18x xÎ3 exÎ3 “ œ 7 327 ' ˆ b54 ‰ 327 Î3 7Î3 œ 7Î3 Ê e e e _ 2 bÄ_ _ x2 7 exÎ3 7; 54 e327 Š 3b eb18b Î3 7Î3 ‹ œ 2 _ 2 dx converges Ê ! ennÎ3 converges n œ7 2 25 36 ! nnÎ3 converges Ê ! ennÎ3 œ e11Î3 e24Î3 e91 e16 4Î3 e5Î3 e2 e n œ7 n œ1 10. faxb œ x2 x 2x4 1 œ axx14b2 is continuous for x decreasing for x œ lim bÄ_ _ _ x4 8; '8 x1 3 1 3 ”'8 ax 1b2 dx '8 ax 1b2 dx• œ b lim ”'8 x 1 dx '8 ax 1b2 dx• bÄ_ Ä_ dx œ lim ax 1 b 2 b ’lnlx 1l x 3 1 “ œ lim 8 _ 2, f is positive for x 4, and f w axb œ ax71xb3 0 for x 7, thus f is bÄ_ b b b _ x4 ˆlnlb 1l b 3 1 ln 7 37 ‰ œ _ Ê ' 8 ax 1 b 2 b dx diverges _ 1 2 3 Ê ! n2 n 2n4 1 diverges Ê ! n2 n 2n4 1 œ 2 41 0 16 25 36 ! n2 n 2n4 1 diverges n œ8 n œ2 " 11. converges; a geometric series with r œ 10 1 n œ8 12. converges; a geometric series with r œ "e 1 n 13. diverges; by the nth-Term Test for Divergence, n lim œ1Á0 Ä _ n1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.3 The Integral Test _ 5 14. diverges by the Integral Test; '1 x 5 1 dx œ 5 ln (n 1) 5 ln 2 Ê '1 n _ _ nœ1 nœ1 x 1 dx Ä _ 15. diverges; ! È3n œ 3 ! È"n , which is a divergent p-series (p œ #" ) _ _ nœ1 nœ1 " 16. converges; ! nÈ2n œ 2 ! n$Î# , which is a convergent p-series (p œ 3# ) 17. converges; a geometric series with r œ "8 1 _ _ _ _ nœ1 nœ1 nœ1 nœ1 18. diverges; ! n8 œ 8 ! 1n and since ! "n diverges, 8 ! 1n diverges _ ln x 19. diverges by the Integral Test: '2 lnxx dx œ "# aln# n ln 2b Ê '2 n x dx Ä _ Ô t œ lndxx × _ b dt œ x Ä 'ln 2 tetÎ2 dt œ lim 2tetÎ2 4etÎ2 ‘ ln 2 bÄ_ Õ dx œ et dt Ø œ lim 2ebÎ2 (b 2) 2eÐln 2ÑÎ2 (ln 2 2)‘ œ _ _ ln x 20. diverges by the Integral Test: '2 Èx dx; bÄ_ 21. converges; a geometric series with r œ 23 1 n n n 5 5 ln 5 ˆ ln 5 ‰ ˆ 54 ‰ œ _ Á 0 22. diverges; n lim œ n lim œ n lim Ä _ 4n 3 Ä _ 4n ln 4 Ä _ ln 4 _ _ nœ0 nœ0 23. diverges; ! n21 œ 2 ! n " 1 , which diverges by the Integral Test 24. diverges by the Integral Test: '1 2xdx 1 œ #" ln (2n 1) Ä _ as n Ä _ n n n 2 2 ln 2 25. diverges; n lim a œ n lim œ n lim œ_Á0 1 Ä_ n Ä _ n1 Ä_ dx 26. diverges by the Integral Test: '1 Èx ˆÈ ; x 1‰ – n Èn 27. diverges; n lim œ n lim Ä _ ln n Ä_ " Š 2È ‹ n Š "n ‹ œ n lim Ä_ Ènb1 du u œ Èx " ' ˆÈn 1‰ ln 2 Ä _ as n Ä _ Ä dx u œ ln 2 du œ Èx — Èn # œ_Á 0 n ˆ1 n" ‰ œ e Á 0 28. diverges; n lim a œ n lim Ä_ n Ä_ 29. diverges; a geometric series with r œ ln"# ¸ 1.44 1 30. converges; a geometric series with r œ ln"3 ¸ 0.91 1 _ 31. converges by the Integral Test: '3 Š "x ‹ u œ ln x dx; ” (ln x) È(ln x)# 1 du œ x" dx • _ Ä 'ln 3 È "# u u 1 du Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 585 586 Chapter 10 Infinite Sequences and Series b œ lim csec" kukd ln 3 œ lim csec" b sec" (ln 3)d œ lim cos" ˆ "b ‰ sec" (ln 3)‘ bÄ_ bÄ_ bÄ_ œ cos" (0) sec" (ln 3) œ 1# sec" (ln 3) ¸ 1.1439 _ 32. converges by the Integral Test: '1 " x a1 ln# xb dx œ '1_ 1 Š(ln‹x) dx; ” u œ ln" x • Ä '0_ 1"u du " x du œ x dx # # œ lim ctan" ud 0 œ lim atan" b tan" 0b œ 1# 0 œ 1# b bÄ_ bÄ_ 33. diverges by the nth-Term Test for divergence; n lim n sin ˆ "n ‰ œ n lim Ä_ Ä_ sin ˆ "n ‰ sin x œ1Á0 ˆ "n ‰ œ xlim Ä0 x 34. diverges by the nth-Term Test for divergence; n lim n tan ˆ "n ‰ œ n lim Ä_ Ä_ Š n"# ‹ sec# ˆ n" ‰ tan ˆ "n ‰ œ lim ˆ "n ‰ nÄ_ Š " ‹ n# œ n lim sec# ˆ "n ‰ œ sec# 0 œ 1 Á 0 Ä_ _ 35. converges by the Integral Test: '1 uœe ex 1 e2x dx; ” du œ ex dx • x Ä 'e _ b " ctan" ud e 1 u# du œ n lim Ä_ œ lim atan" b tan" eb œ 1# tan" e ¸ 0.35 bÄ_ _ 36. converges by the Integral Test: '1 u œ ex × _ _ 2 du œ ex dx Ä ' u(1 2 u) du œ ' ˆ u2 u 2 1 ‰ du 1 ex dx; e e Õ dx œ " du Ø u Ô b œ lim 2 ln u u 1 ‘ e œ lim 2 ln ˆ b b 1 ‰ 2 ln ˆ e e 1 ‰ œ 2 ln 1 2 ln ˆ e e 1 ‰ œ 2 ln ˆ e e 1 ‰ ¸ 0.63 bÄ_ bÄ_ _ 8 tanc" x 37. converges by the Integral Test: '1 1 x# _ x 38. diverges by the Integral Test: '1 dx; ” u œ tan" x '11ÎÎ42 8u du œ c4u# d 11ÎÎ24 œ 4 Š 14# 161# ‹ œ 341# dx • Ä du œ 1 x# u œ x# 1 x# 1 dx; ” du œ 2x dx • _ 39. converges by the Integral Test: '1 sech x dx œ 2 lim _ du Ä #" '2 4 b œ lim #" ln u‘ 2 œ lim " (ln b ln 2) œ _ bÄ_ # bÄ_ '1b 1 eae b dx œ 2 lim ctan" ex d b1 x x # bÄ_ œ 2 lim atan" eb tan" eb œ 1 2 tan" e ¸ 0.71 bÄ_ bÄ_ _ 40. converges by the Integral Test: '1 sech# x dx œ lim bÄ_ œ 1 tanh 1 ¸ 0.76 _ '1b sech# x dx œ lim ctanh xd b1 œ lim (tanh b tanh 1) bÄ_ bÄ_ 41. '1 ˆ x a 2 x " 4 ‰ dx œ lim ca ln kx 2k ln kx 4kd 1 œ lim ln (bb2)4 ln ˆ 35 ‰ ; a b bÄ_ a _, a 1 lim (bb2)4 œ a lim (b 2)a1 œ œ 1, a œ 1 bÄ_ bÄ_ a bÄ_ Ê the series converges to ln ˆ 53 ‰ if a œ 1 and diverges to _ if a 1. If a 1, the terms of the series eventually become negative and the Integral Test does not apply. From that point on, however, the series behaves like a negative multiple of the harmonic series, and so it diverges. _ x1 b1 ‰ ˆ2‰ 42. '3 ˆ x " 1 x 2a 1 dx œ lim ’ln ¹ (x 1)2a ¹“ œ lim ln (b 1)2a ln 42a ; lim b bÄ_ 1, a œ "# " œ 2a c 1 _, a " b Ä _ #a(b 1) # œ lim 3 bÄ_ b" 2a b Ä _ (b 1) Ê the series converges to ln ˆ #4 ‰ œ ln 2 if a œ #" and diverges to _ if Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.3 The Integral Test 587 if a "# . If a "# , the terms of the series eventually become negative and the Integral Test does not apply. From that point on, however, the series behaves like a negative multiple of the harmonic series, and so it diverges. 43. (a) (b) There are (13)(365)(24)(60)(60) a10* b seconds in 13 billion years; by part (a) sn Ÿ 1 ln n where n œ (13)(365)(24)(60)(60) a10* b Ê sn Ÿ 1 ln a(13)(365)(24)(60)(60) a10* bb œ 1 ln (13) ln (365) ln (24) 2 ln (60) 9 ln (10) ¸ 41.55 _ _ _ nœ1 nœ1 nœ1 " 44. No, because ! nx œ x" ! n" and ! n" diverges _ _ _ nœ1 nœ1 nœ1 45. Yes. If ! an is a divergent series of positive numbers, then ˆ "# ‰ ! an œ ! ˆ a#n ‰ also diverges and a#n an . _ There is no “smallest" divergent series of positive numbers: for any divergent series ! an of positive numbers nœ1 _ ! ˆ an ‰ has smaller terms and still diverges. nœ1 # _ _ _ nœ1 nœ1 nœ1 46. No, if ! an is a convergent series of positive numbers, then 2 ! an œ ! 2an also converges, and 2an an . There is no “largest" convergent series of positive numbers. 47. (a) Both integrals can represent the area under the curve faxb œ Èx1 1 , and the sum s50 can be considered an 50 approximation of either integral using rectangles with ?x œ 1. The sum s50 œ ! Èn1 1 is an overestimate of the nœ1 51 integral '1 Èx1 1 dx. The sum s50 represents a left-hand sum (that is, the we are choosing the left-hand endpoint of each subinterval for ci ) and because f is a decreasing function, the value of f is a maximum at the left-hand endpoint of each sub interval. The area of each rectangle overestimates the true area, thus '1 51 50 1 ! 1 . In a similar Èx 1 dx Èn 1 nœ1 50 manner, s50 underestimates the integral '0 Èx1 1 dx. In this case, the sum s50 represents a right-hand sum and because f is a decreasing function, the value of f is aminimum at the right-hand endpoint of each subinterval. The area of each rectangle underestimates the true area, thus ! Èn1 1 '0 50 50 nœ1 œ ’2Èx 1“ œ 2È52 2È2 ¸ 11.6 and '0 51 50 1 1 Èx 1 dx. Evaluating the integrals we find 50 1 È Èx 1 dx œ ’2 x 1“ 0 '151 Èx1 1 dx œ 2È51 2È1 ¸ 12.3. Thus, 50 11.6 ! Èn1 1 12.3. nœ1 nb1 (b) sn 1000 Ê '1 Ên 2 nb1 1 È œ 2Èn 1 2È2 1000 Ê n Š500 2È2‹ Èx 1 dx œ ’2 x 1“ 1 251415. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. ¸ 251414.2 588 Chapter 10 Infinite Sequences and Series 30 _ nœ1 nœ1 _ 48. (a) Since we are using s30 œ ! n14 to estimate ! n14 , the error is given by ! n14 . We can consider this sum as an estimate nœ31 of the area under the curve faxb œ x14 when x 30 using rectangles with ?x œ 1 and ci is the right-hand endpoint of each subinterval. Since f is a decreasing function, the value of f is a minimum at the right-hand endpoint of each 1 subinterval, thus ! n14 '30 x14 dx œ lim '30 x14 dx œ lim ’ 3x1 3 “ œ lim Š 3b1 3 3a30 ‹ ¸ 1.23 ‚ 105 . b3 _ _ b b bÄ_ nœ31 bÄ_ bÄ_ 30 5 Thus the error 1.23 ‚ 10 Þ (b) We want S sn 0.000001 Ê 'n _ œ 1 x4 dx 0.000001 Ê 4 _ 1 x3 dx 0.01 Ê œ 2n1 2 0.01 Ê n È50 ¸ 7.071 Ê n 4 bÄ_ 3 1000000 lim ˆ 3b1 3 3n1 3 ‰ œ 3n1 3 0.000001 Ê n É ¸ 69.336 Ê n 3 bÄ_ 49. We want S sn 0.01 Ê 'n 3 bÄ_ n 70. 'n_ x1 dx œ lim 'nb x1 dx œ lim ’ 2x1 “b œ lim ˆ 2b1 2n1 ‰ 3 bÄ_ 3 bÄ_ 2 2 bÄ_ n 2 8 8 Ê S ¸ s8 œ ! n13 ¸ 1.195 nœ1 50. We want S sn 0.1 Ê 'n ' _ œ 'n_ x1 dx œ lim 'nb x1 dx œ lim ’ 3x1 “b b b 1 1 1 1 ˆ x ‰ lim tan ’ “ 2 4 dx œ x2 4 dx 0.1 Ê blim x 2 2 Ä_ n bÄ_ n 1 1 1 1 1 1 ˆ b ‰ 1 ˆ n ‰ ‰ 1 ˆ n ‰ ˆ ˆ ‰ lim tan œ 4 2 tan 2 2 tan 2 2 0.1 Ê n 2tan 2 0.2 ¸ 9.867 Ê n bÄ_ 2 10 Ê S ¸ s10 10 œ ! n2 1 4 ¸ 0.57 nœ1 51. S sn 0.00001 Ê 'n _ 1 x1.1 dx 0.00001 Ê 'n_ x1 dx œ lim 'nb x1 dx œ lim ’ x10 “b œ lim ˆ b10 n10 ‰ 1.1 bÄ_ 1.1 bÄ_ 0.1 bÄ_ n 0.1 0.1 10 œ n10 Ê n 1060 0.1 0.00001 Ê n 1000000 52. S sn 0.01 Ê 'n _ œ 1 dx 0.01 Ê xaln xb3 'n_ xaln1xb dx œ lim 'nb xaln1xb dx œ lim ’ 2aln1xb “b 3 bÄ_ 3 È lim Š 2aln1bb2 2aln1nb2 ‹ œ 2aln1nb2 0.01 Ê n e 50 ¸ 1177.405 Ê n bÄ_ n n kœ1 kœ1 bÄ_ 2 n 1178 53. Let An œ ! ak and Bn œ ! 2k aa2k b , where {ak } is a nonincreasing sequence of positive terms converging to 0. Note that {An } and {Bn } are nondecreasing sequences of positive terms. Now, Bn œ 2a# 4a% 8a) á 2n aa2n b œ 2a# a2a% 2a% b a2a) 2a) 2a) 2a) b á ˆ2aa2n b 2aa2n b á 2aa2n b ‰ Ÿ 2a" 2a# a2a$ 2a% b a2a& 2a' 2a( 2a) b á ðóóóóóóóóóóóóóóñóóóóóóóóóóóóóóò 2n1 terms _ ˆ2aa2nc1 b 2aa2nc1 1b á 2aa2n b ‰ œ 2Aa2n b Ÿ 2 ! ak . Therefore if ! ak converges, kœ1 then {Bn } is bounded above Ê ! 2k aa2k b converges. Conversely, _ An œ a" aa# a$ b aa% a& a' a( b á an a" 2a# 4a% á 2n aa2n b œ a" Bn a" ! 2k aa2k b . kœ1 _ Therefore, if ! 2 aa2k b converges, then {An } is bounded above and hence converges. k kœ1 _ _ _ nœ2 nœ2 nœ2 " " ! " ! 2 n a a2 n b œ ! 2 n n " 54. (a) aa2n b œ 2n ln"a2n b œ 2n †n(ln # †n(ln 2) œ ln # 2) Ê n , which diverges _ Ê ! n ln" n diverges. nœ2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.3 The Integral Test _ _ _ _ nœ1 nœ1 nœ1 nœ1 589 n (b) aa2n b œ #"np Ê ! 2n aa2n b œ ! 2n † #"np œ ! a2n"bpc1 œ ! ˆ #p"c1 ‰ , a geometric series that converges if #p"c1 1 or p 1, but diverges if p Ÿ 1. _ 55. (a) '2 '_ u œ ln x cpb1 dx ucp du œ lim ’ up 1 “ œ lim Š 1 " p ‹ cbp1 (ln 2)p1 d x(ln x)p ; ” du œ dx • Ä ln 2 b Ä _ bÄ_ ln 2 x " c pb1 ,p1 p 1 (ln 2) Ê the improper integral converges if p 1 and diverges if p 1. œœ b _, p " _ dx For p œ 1: '2 p œ 1. cln (ln x)d 2 œ lim cln (ln b) ln (ln 2)d œ _, so the improper integral diverges if x ln x œ b lim Ä_ bÄ_ b _ (b) Since the series and the integral converge or diverge together, ! n(ln" n)p converges if and only if p 1. nœ2 56. (a) p œ 1 Ê the series diverges (b) p œ 1.01 Ê the series converges _ _ nœ2 nœ2 (c) ! n aln" n$ b œ 3" ! n(ln" n) ; p œ 1 Ê the series diverges (d) p œ 3 Ê the series converges nb1 57. (a) From Fig. 10.11(a) in the text with f(x) œ "x and ak œ k" , we have '1 " " " " x dx Ÿ 1 # 3 á n Ÿ 1 '1 f(x) dx Ê ln (n 1) Ÿ 1 "# 3" á n" Ÿ 1 ln n Ê 0 Ÿ ln (n 1) ln n n Ÿ ˆ1 "# 3" á n" ‰ ln n Ÿ 1. Therefore the sequence ˜ˆ1 #" 3" á n" ‰ ln n™ is bounded above by 1 and below by 0. nb1 (b) From the graph in Fig. 10.11(b) with f(x) œ "x , n " 1 'n " x dx œ ln (n 1) ln n Ê 0 n " 1 cln (n 1) ln nd œ ˆ1 #" 3" á n" 1 ln (n 1)‰ ˆ1 #" 3" á n" ln n‰ . If we define an œ 1 "# œ 3" n" ln n, then 0 an1 an Ê an1 an Ê {an } is a decreasing sequence of nonnegative terms. # 58. ex Ÿ ex for x _ _ # b 1, and '1 ecx dx œ lim cex d " œ lim êb e1 ‰ œ ec1 Ê '1 ecx dx converges by bÄ_ bÄ_ _ n # the Comparison Test for improper integrals Ê ! e nœ0 _ 59. (a) s10 œ ! n"3 œ 1.97531986; '11 x13 dx œ lim 10 bÄ_ nœ1 _ # œ 1 ! en converges by the Integral Test. nœ1 1 ‰ 1 '11b x3 dx œ lim ’ x2c “ b œ lim ˆ 2b1 242 œ 242 and 2 bÄ_ 11 1 ‰ 1 '10_ x1 dx œ lim '10b x3 dx œ lim ’ x2c “ b œ lim ˆ 2b1 200 œ 200 bÄ_ 2 2 3 (b) bÄ_ bÄ_ 10 bÄ_ 2 1 1 Ê 1.97531986 242 s 1.97531986 200 Ê 1.20166 s 1.20253 _ s œ ! n"3 ¸ 1.20166 2 1.20253 œ 1.202095; error Ÿ 1.20253 2 1.20166 œ 0.000435 nœ1 _ 60. (a) s10 œ ! n"4 œ 1.082036583; '11 x14 dx œ lim 10 bÄ_ nœ1 1 ‰ 1 '11b x4 dx œ lim ’ x3c “ b œ lim ˆ 3b1 3993 œ 3993 and 3 bÄ_ 11 1 ‰ 1 '10_ x1 dx œ lim '10b x4 dx œ lim ’ x3c “ b œ lim ˆ 3b1 3000 œ 3000 bÄ_ 3 4 bÄ_ bÄ_ 10 bÄ_ 3 1 1 Ê 1.082036583 3993 s 1.082036583 3000 Ê 1.08229 s 1.08237 _ (b) s œ ! n"4 ¸ 1.08229 2 1.08237 œ 1.08233; error Ÿ 1.08237 2 1.08229 œ 0.00004 nœ1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 3 590 Chapter 10 Infinite Sequences and Series 10.4 COMPARISON TESTS _ 1. Compare with ! n"2 , which is a convergent p-series, since p œ 2 1. Both series have nonnegative terms for n 1. For nœ1 n 1, we have n2 Ÿ n2 30 Ê n12 1 n2 30 . _ Then by Comparison Test, ! n2 1 30 converges. nœ1 _ 2. Compare with ! n"3 , which is a convergent p-series, since p œ 3 1. Both series have nonnegative terms for n 1. For nœ1 n 1, we have n4 Ÿ n4 2 Ê n14 1 n n 4 2 Ê n 4 n 1 n 4 2 Ê n 3 n n 4 2 _ n1 n 4 2 . Then by Comparison Test, ! nn412 nœ1 converges. _ 3. Compare with ! È"n , which is a divergent p-series, since p œ #" Ÿ 1. Both series have nonnegative terms for n 2. For nœ2 n 2, we have Èn 1 Ÿ Èn Ê Èn1 1 1 Èn . _ Then by Comparison Test, ! Èn1 1 diverges. nœ2 _ 4. Compare with ! "n , which is a divergent p-series, since p œ 1 Ÿ 1. Both series have nonnegative terms for n 2. For nœ2 n 2, we have n2 n Ÿ n2 Ê n2 1 n n 1 n2 n2 œ n Ê n2 n 1 n n2 Ê n2 n n 1 n. n2 n _ Thus ! nn22n diverges. nœ2 _ 5. Compare with ! n3"Î2 , which is a convergent p-series, since p œ 32 1. Both series have nonnegative terms for n 1. nœ1 _ 2 2 n n 1, we have 0 Ÿ cos2 n Ÿ 1 Ê cos Ÿ n31Î2 . Then by Comparison Test, ! cos converges. n3Î2 n3Î2 For n nœ1 _ 6. Compare with ! 3"n , which is a convergent geometric series, since lrl œ ¹ 13 ¹ 1. Both series have nonnegative terms for nœ1 n 1. For n _ 1, we have n † 3n 3n Ê n†13n Ÿ 31n . Then by Comparison Test, ! n†13n converges. _ _ _ nœ1 nœ1 È nœ1 È 7. Compare with ! n3Î52 . The series ! n31Î2 is a convergent p-series, since p œ 23 1, and the series ! n3Î52 nœ1 _ œ È5 ! n31Î2 converges by Theorem 8 part 3. Both series have nonnegative terms for n 1. For n 1, we have nœ1 n3 Ÿ n4 Ê 4n3 Ÿ 4n4 Ê n4 4n3 Ÿ n4 4n4 œ 5n4 Ê n4 4n3 Ÿ 5n4 20 œ 5an4 4b Ê nn44n4 Ÿ 5. 4 3 _ È Ê n na4n44b Ÿ 5 Ê nn444 Ÿ n53 Ê É nn444 Ÿ É n53 œ n3Î52 Then by Comparison Test, ! É nn444 converges. 3 nœ1 _ 8. Compare with ! È"n , which is a divergent p-series, since p œ #" Ÿ 1. Both series have nonnegative terms for n nœ1 n 1, we have Èn Ê n2 2 nÈn n Èn 1 ÊÈ2 n 3 1 Èn . 1 Ê 2Èn n2 3 Ê 2 Ê 2Èn 1 n ˆn 2 È n 1 ‰ n2 3 3 Ê nˆ2Èn 1‰ 1Ê _ Èn 1 Then by Comparison Test, ! È 2 nœ1 n 2È n 1 n2 3 n 3 3 Ê 2 nÈn n 3n ˆÈ n 1 ‰ 1 n Ê n2 3 2 ˆÈ ‰ 1 Ê nn2 31 n Ê diverges. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 3 2 É 1n 1. For Section 10.4 Comparison Tests _ 9. Compare with ! n"2 , which is a convergent p-series, since p œ 2 1. Both series have positive terms for n nc2 n3 c n2 b 3 1 În 2 œ lim _ ! nœ1 nÄ_ nœ1 591 1. lim bann nÄ_ 3n 4n 6n 4 6 œ lim n3nn22n 3 œ lim 3n lim 6n 2 2n œ 2 œ lim 6 œ 1 0. Then by Limit Comparison Test, 3 2 2 nÄ_ nÄ_ nÄ_ nÄ_ n2 n3 n2 3 converges. _ 10. Compare with ! È"n , which is a divergent p-series, since p œ #" Ÿ 1. Both series have positive terms for n É nn2bb12 nœ1 1. lim bann nÄ_ n n n 2n 1 2 È1 œ 1 0. Then by Limit Comparison œ lim 1ÎÈn œ lim É nn2 2n œ É lim 2 œ 2 œ É lim n2 2 œ É lim 2 nÄ_ 2 nÄ_ nÄ_ nÄ_ nÄ_ _ Test, ! É nn212 diverges. nœ1 _ 11. Compare with ! "n , which is a divergent p-series, since p œ 1 Ÿ 1. Both series have positive terms for n nan b 1b Šn2 œ lim nœ2 b 1‹an c 1b 3 2 2 nÄ_ _ Test, nÄ_ 2n 6n 2 6 œ lim n3 nn2+nn 1 œ lim 3n3n lim 6n 2 2n 1 œ 2 œ lim 6 œ 1 0. Then by Limit Comparison 1 În nÄ_ 2. lim bann nÄ_ nÄ_ nÄ_ ! 2 nan 1b diverges. an 1ban 1b nœ2 _ 12. Compare with ! 2"n , which is a convergent geometric series, since lrl œ ¹ 21 ¹ 1. Both series have positive terms for nœ1 2n n lim an œ lim 13 bÎ24nn nÄ_ bn nÄ_ 1. n _ n n ln 4 ! 2 n converges. œ lim 3 4 4n œ lim 44n ln 4 œ 1 0. Then by Limit Comparison Test, 34 nÄ_ nÄ_ nœ1 _ 13. Compare with ! È"n , which is a divergent p-series, since p œ 12 Ÿ 1. Both series have positive terms for n nœ1 5n n†4n È œ lim 1ÎÈn œ lim 54n œ n nÄ_ nÄ_ _ n lim ˆ 5 ‰ œ nÄ_ 4 _ 1. lim bann nÄ_ n _. Then by Limit Comparison Test, ! È5n†4n diverges. nœ1 n 14. Compare with ! ˆ 25 ‰ , which is a convergent geometric series, since lrl œ ¹ 25 ¹ 1. Both series have positive terms for nœ1 ˆ 2n b 3 ‰ 15 ‰n 15 ‰n 15 ‰ n 1. lim bann œ lim a5n2Îb54bn œ lim ˆ 10n œ exp lim lnˆ 10n œ exp lim n lnˆ 10n 10n 8 10n 8 nÄ_ nÄ_ nÄ_ 10n 8 nÄ_ nÄ_ b 15 ‰ 10 lnˆ 10n 10b 8 70n2 70n2 10n b 8 œ exp lim œ exp lim 10n b151În10n œ exp lim a10n 15 2 2 1 În ba10n 8b œ exp nlim nÄ_ nÄ_ nÄ_ Ä_ 100n 230n 120 n _ n 3‰ 140 7Î10 œ exp lim 200n140n 0. Then by Limit Comparison Test, ! ˆ 2n converges. 230 œ exp lim 200 œ e 5n 4 nÄ_ nÄ_ nœ1 _ 15. Compare with ! "n , which is a divergent p-series, since p œ 1 Ÿ 1. Both series have positive terms for n nœ2 2. lim bann nÄ_ _ " ln n œ lim 1În œ lim lnnn œ lim 11În œ lim n œ _. Then by Limit Comparison Test, ! ln"n diverges. nÄ_ nÄ_ nÄ_ nÄ_ nœ2 _ 16. Compare with ! n"2 , which is a convergent p-series, since p œ 2 1. Both series have positive terms for n nœ1 œ lim nÄ_ lnŠ1 n"2 ‹ 1 În 2 2 " Š n3 ‹ 1 n2 Š n23 ‹ 1 œ lim nÄ_ _ 1. lim bann nÄ_ œ lim 1 1 " œ 1 0. Then by Limit Comparison Test, ! lnˆ1 n"2 ‰ converges. nÄ_ n2 nœ1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 592 Chapter 10 Infinite Sequences and Series _ 17. diverges by the Limit Comparison Test (part 1) when compared with ! È"n , a divergent p-series: nœ1 " lim nÄ_ Œ #Èn È $ n Š " ‹ Èn Èn œ n lim lim ˆ " ‰ œ #" $ n œ Ä _ 2È n È n Ä _ # n 1Î6 18. diverges by the Direct Comparison Test since n n n n Èn 0 Ê n 3Èn "n , which is the nth _ term of the divergent series ! "n or use Limit Comparison Test with bn œ n" nœ1 # 19. converges by the Direct Comparison Test; sin2n n Ÿ #"n , which is the nth term of a convergent geometric series n 20. converges by the Direct Comparison Test; 1 ncos Ÿ n2# and the p-series ! n"# converges # 2n 21. diverges since n lim œ 23 Á 0 Ä _ 3n 1 " 22. converges by the Limit Comparison Test (part 1) with n$Î# , the nth term of a convergent p-series: lim nÄ_ Š nn# È"n ‹ ˆ n n " ‰ œ 1 œ n lim Ä_ " ‹ Š $Î# n 23. converges by the Limit Comparison Test (part 1) with n"# , the nth term of a convergent p-series: lim nÄ_ Š n(n 10n1)(n" 2) ‹ Š n"# ‹ # 10n n 20n 1 20 œ n lim œ n lim œ n lim œ 10 Ä _ n# 3n 2 Ä _ 2n 3 Ä_ 2 24. converges by the Limit Comparison Test (part 1) with n"# , the nth term of a convergent p-series: lim n# (n 5n$ 3n 2) Šn# 5‹ Š n"# ‹ nÄ_ $ # 5n 3n 15n 3 30n œ n lim œ n lim œ n lim œ5 Ä _ n$ 2n# 5n 10 Ä _ 3n# 4n 5 Ä _ 6n 4 n n n n ‰ 25. converges by the Direct Comparison Test; ˆ 3n n 1 ‰ ˆ 3n œ ˆ "3 ‰ , the nth term of a convergent geometric series " 26. converges by the Limit Comparison Test (part 1) with n$Î# , the nth term of a convergent p-series: lim nÄ_ " Š $Î# ‹ $ n Š È $" n 2 ‹ É n n$ 2 œ lim É1 n2$ œ 1 œ n lim Ä_ nÄ_ _ " ! " diverges 27. diverges by the Direct Comparison Test; n ln n Ê ln n ln ln n Ê "n ln"n ln (ln n) and n nœ3 _ 28. converges by the Limit Comparison Test (part 2) when compared with ! n"# , a convergent p-series: nœ1 # lim nÄ_ ’ (lnn$n) “ Š n"# ‹ # (ln n) œ n lim œ n lim n Ä_ Ä_ 2(ln n) Š n" ‹ 1 ln n œ 2 n lim œ0 Ä_ n 29. diverges by the Limit Comparison Test (part 3) with "n , the nth term of the divergent harmonic series: lim nÄ_ ’ Èn1ln n “ ˆ n" ‰ Èn œ n lim œ n lim Ä _ ln n Ä_ " Š 2È ‹ n ˆ n" ‰ œ n lim Ä_ Èn 2 œ_ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.4 Comparison Tests 593 " 30. converges by the Limit Comparison Test (part 2) with n&Î% , the nth term of a convergent p-series: # lim nÄ_ n) ’ (ln$Î# “ n " Š &Î% ‹ n ˆ 2 lnn n ‰ # (ln n) œ n lim œ n lim Ä _ n"Î% Ä_ " Š $Î% ‹ 4n ln n œ 8 n lim œ 8 n lim Ä _ n"Î% Ä_ ˆ n" ‰ Š " ‹ 4n$Î% " œ 32 n lÄ im_ n"Î% œ 32 † 0 œ 0 31. diverges by the Limit Comparison Test (part 3) with "n , the nth term of the divergent harmonic series: lim nÄ_ ˆ 1 "ln n ‰ n " œ n lim œ n lim nœ_ ˆ "n ‰ œ n lim Ä _ 1 ln n Ä _ Š"‹ Ä_ n _ ln (x 1) 32. diverges by the Integral Test: '2 x1 _ b dx œ 'ln 3 u du œ lim "2 u# ‘ ln 3 œ lim " # ab bÄ_ # bÄ_ ln# 3b œ _ " 33. converges by the Direct Comparison Test with n$Î# , the nth term of a convergent p-series: n# 1 n for " " n 2 Ê n# an# 1b n$ Ê nÈn# 1 n$Î# Ê $Î# or use Limit Comparison Test with 1# . nÈ n# 1 n n " 34. converges by the Direct Comparison Test with n$Î# , the nth term of a convergent p-series: n# 1 n# Èn # " " Ê n# 1 Ènn$Î# Ê nÈn1 n$Î# Ê n# 1 n$Î# or use Limit Comparison Test with n$Î# . _ _ _ nœ1 nœ1 nœ1 35. converges because ! "n2nn œ ! n2" n ! " #n which is the sum of two convergent series: _ ! nœ1 _ " " " ! "n is a convergent geometric series n2n converges by the Direct Comparison Test since n#n #n , and 2 nœ1 _ _ 36. converges by the Direct Comparison Test: ! nn# 22n œ ! ˆ n2" n n"# ‰ and n2" n n"# Ÿ #"n n"# , the sum of n nœ1 nœ1 the nth terms of a convergent geometric series and a convergent p-series 37. converges by the Direct Comparison Test: 3nc1" 1 3n"c1 , which is the nth term of a convergent geometric series nc1 ˆ " 3"n ‰ œ "3 Á 0 38. diverges; n lim Š 3 3n" ‹ œ n lim Ä_ Ä_ 3 _ n 39. converges by Limit Comparison Test: compare with ! ˆ 15 ‰ , which is a convergent geometric series with lrl œ 15 1, nœ1 lim nÄ_ 1 1 Š n2n b b 3n † 5n ‹ a 1 Î5 b n n1 1 œ n lim œ n lim œ 0. Ä _ n2 3n Ä _ 2n 3 _ n 40. converges by Limit Comparison Test: compare with ! ˆ 34 ‰ , which is a convergent geometric series with lrl œ 15 1, nœ1 3 Š 23n b b 4n ‹ n n lim n n Ä _ a 3 Î4 b 8 ‰n ˆ 12 1 8 12 œ n lim lim œ 11 œ 1 0. n n 12n œ 9 ˆ Ä_ n Ä _ 129 ‰ 1 n n Š 2n 2cnn ‹ n _ 41. diverges by Limit Comparison Test: compare with ! 1n , which is a divergent p-series, n lim Ä_ nœ1 † 1În 2 n œ n lim Ä _ 2n n 2n aln 2b2 2n ln 2 1 œ n lim œ n lim œ 1 0. Ä _ 2n ln 2 Ä _ 2n aln 2b2 _ _ nœ1 nœ1 42. diverges by the definition of an infinite series: ! lnˆ n n 1 ‰ œ ! ln n ln an 1b‘, sk œ aln 1 ln 2b aln 2 ln 3b Þ Þ Þ alnak 1b ln kb aln k ln ak 1bb œ ln ak 1b Ê lim sk œ _ kÄ_ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 594 Chapter 10 Infinite Sequences and Series _ _ _ nœ2 nœ2 nœ2 43. converges by Comparison Test with ! nan 1 1b which converges since ! nan 1 1b œ ! ’ n 1 1 n1 “, and sk œ ˆ1 12 ‰ ˆ 12 13 ‰ Þ Þ Þ ˆ k 1 2 k 1 1 ‰ ˆ k 1 1 1k ‰ œ 1 1k Ê lim sk œ 1; for n Ê nan 1ban 2b! nan 1b Ê n! 2, an 2b! kÄ_ nan 1b Ê n!1 Ÿ nan11b _ an c 1bx n 2x 44. converges by Limit Comparison Test: compare with ! n13 , which is a convergent p-series, n lim Ä _ 1În3 a b nœ1 n a n 1 bx n 2n 2 œ n lim œ n lim œ n lim œ n lim œ10 Ä _ an 2ban 1bnan 1bx Ä _ n2 3n 2 Ä _ 2n 3 Ä_ 2 3 2 45. diverges by the Limit Comparison Test (part 1) with "n , the nth term of the divergent harmonic series: lim nÄ_ ˆsin "n ‰ sin x œ1 ˆ "n ‰ œ xlim Ä0 x 46. diverges by the Limit Comparison Test (part 1) with "n , the nth term of the divergent harmonic series: lim nÄ_ ˆtan "n ‰ ˆsin " ‰ Š " ‹ ˆ " ‰n œ lim ˆ cos" x ‰ ˆ sinx x ‰ œ 1 † 1 œ 1 ˆ "n ‰ œ n lim Ä _ cos " xÄ0 n n c" 1 _ _ 1 47. converges by the Direct Comparison Test: tann1.1 n n#1.1 and ! n#1.1 œ 1# ! n"1.1 is the product of a nœ1 nœ1 convergent p-series and a nonzero constant c" ˆ1‰ Þ Þ _ 48. converges by the Direct Comparison Test: sec" n 1# Ê secn1 3 n n#1 3 and ! nœ1 _ ˆ 1# ‰ 1 ! " n1 3 œ # n1 3 is the Þ Þ nœ1 product of a convergent p-series and a nonzero constant 49. converges by the Limit Comparison Test (part 1) with n"# : n lim Ä_ n Š coth ‹ n# Š n"# ‹ c2n cn e e œ n lim coth n œ n lim Ä_ Ä _ en ecn n "e œ n lim œ1 Ä _ 1 ec2n 50. converges by the Limit Comparison Test (part 1) with n"# : n lim Ä_ n Š tanh ‹ n# Š n"# ‹ c2n e e œ n lim tanh n œ n lim Ä_ Ä _ en e n n n "e œ n lim œ1 Ä _ 1 ec2n 51. diverges by the Limit Comparison Test (part 1) with 1n : n lim Ä_ 52. 1 Š nÈ n n‹ ˆ 1n ‰ converges by the Limit Comparison Test (part 1) with n"# : n lim Ä_ 1 œ n lim n n œ 1. Ä_ È Èn n Š n# ‹ Š n"# ‹ n È œ n lim nœ1 Ä_ 53. 1 2 3" á n œ ˆ n(n" 1) ‰ œ n(n 2 1) . The series converges by the Limit Comparison Test (part 1) with n"# : # lim nÄ_ Š nan 2b 1b ‹ Š n"# ‹ # 2n 4n 4 œ n lim œ n lim œ n lim œ 2. Ä _ n# n Ä _ 2n 1 Ä_ 2 " 6 6 54. 1 2# 3"# á n# œ n(n b 1)(2n b 1) œ n(n 1)(2n 1) Ÿ n$ Ê the series converges by the Direct Comparison Test 6 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 1 Section 10.4 Comparison Tests 595 an 55. (a) If n lim œ 0, then there exists an integer N such that for all n N, ¹ bann 0¹ 1 Ê 1 bann 1 Ä _ bn Ê an bn . Thus, if ! bn converges, then ! an converges by the Direct Comparison Test. an (b) If n lim œ _, then there exists an integer N such that for all n N, bann 1 Ê an bn . Thus, if Ä _ bn ! bn diverges, then ! an diverges by the Direct Comparison Test. _ 56. Yes, ! ann converges by the Direct Comparison Test because ann an nœ1 an 57. n lim œ _ Ê there exists an integer N such that for all n N, bann 1 Ê an bn . If ! an converges, Ä _ bn then ! bn converges by the Direct Comparison Test 58. ! an converges Ê n lim a œ 0 Ê there exists an integer N such that for all n N, 0 Ÿ an 1 Ê an# an Ä_ n Ê ! a#n converges by the Direct Comparison Test 59. Since an 0 and n lim a œ _ Á 0, by nth term test for divergence, ! an diverges. Ä_ n an 60. Since an 0 and n lim a n2 † an b œ 0, compare !an with ! n"# , which is a convergent p-series; n lim Ä_ Ä _ 1În2 œ n lim a n2 † an b œ 0 Ê !an converges by Limit Comparison Test Ä_ _ q _ 61. Let _ q _ and p 1. If q œ 0, then ! alnnnp b œ ! n1p , which is a convergent p-series. If q Á 0, compare with nœ2 _ ! nœ2 1 nr where 1 r p, then n lim Ä_ aln nbq np 1Înr nœ2 aln nbq aln nbq œ n lim , and p r 0. If q 0 Ê q 0 and n lim Ä _ npcr Ä _ npcr qaln nbqc1 ˆ 1 ‰ q qc1 qaln nb aln nb 1 n œ n lim œ 0. If q 0, n lim œ n lim œ n lim . If q 1 Ÿ 0 Ê 1 q cq Ä _ aln nb npcr Ä _ npcr Ä _ ap rbnpcrc1 Ä _ ap rbnpcr lim qaln nbqc1 n Ä _ ap rbnpcr q œ n lim œ 0, otherwise, we apply L'Hopital's Rule again. n lim Ä _ ap rbnpcr aln nb1cq Ä_ qc2 qaq 1baln nb œ n lim . If q 2 Ÿ 0 Ê 2 q Ä _ ap rb2 npcr qc2 qaq 1baln nb 0 and n lim Ä _ ap rb2 npcr 0 and qaq 1baln nbqc2 ˆ 1n ‰ ap rb2 npcrc1 q aq 1 b œ n lim œ 0; otherwise, we Ä _ ap rb2 npcr aln nb2cq apply L'Hopital's Rule again. Since q is finite, there is a positive integer k such that q k Ÿ 0 Ê k q 0. Thus, after k qaq 1bâaq k 1baln nbqck qaq 1bâaq k 1b applications of L'Hopital's Rule we obtain n lim œ n lim œ 0. Since the limit is Ä_ Ä _ ap rbk npcr aln nbkcq ap rbk npcr _ q 0 in every case, by Limit Comparison Test, the series ! alnnnpb converges. n œ1 _ q _ 62. Let _ q _ and p Ÿ 1. If q œ 0, then ! alnnnp b œ ! n1p , which is a divergent p-series. If q 0, compare with nœ2 _ ! nœ2 1 np , which is a divergent p-series. Then n lim Ä_ where 0 p r Ÿ 1. n lim Ä_ lim ar p b n rcpc1 n Ä _ aqbaln nbcqc1 ˆ 1n ‰ aln nbq np 1 În r nœ2 aln nbq np 1Înp _ œ n lim aln nbq œ _. If q 0 Ê q 0, compare with ! n1r , Ä_ nœ2 aln nbq nrcp œ n lim œ n lim cq since r p 0. Apply L'Hopital's to obtain Ä _ npcr Ä _ aln nb rcp ar p bn œ n lim . If q 1 Ÿ 0 Ê q 1 Ä _ aqbaln nbcqc1 rcp ar pbn aln nb 0 and n lim a q b Ä_ qb1 2 rcpc1 œ _, 2 rcp a r pb n a r pb n otherwise, we apply L'Hopital's Rule again to obtain n lim œ n lim . If Ä _ aqbaq 1baln nbcqc2 ˆ 1 ‰ Ä _ aqbaq 1baln nbcqc2 n q 2 Ÿ 0 Ê q 2 a r pb2 nrcp a r pb2 nrcp aln nbqb2 0 and n lim œ n lim œ _, otherwise, we aqbaq 1b Ä _ aqbaq 1baln nbcqc2 Ä_ apply L'Hopital's Rule again. Since q is finite, there is a positive integer k such that q k Ÿ 0 Ê q k k rcp k rcp 0. Thus, after qbk a r pb n a r pb n aln nb k applications of L'Hopital's Rule we obtain n lim œ n lim œ _. Ä _ aqbaq 1bâaq k 1baln nbcqck Ä _ aqbaq 1bâaq k 1b Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 596 Chapter 10 Infinite Sequences and Series _ q Since the limit is _ if q 0 or if q 0 and p 1, by Limit comparison test, the series ! alnnpncbr diverges. Finally if q 0 _ q _ _ q n œ1 and p œ 1 then ! alnnnp b œ ! alnnnb . Compare with ! 1n , which is a divergent p-series. For n nœ2 Ê aln nbq _ nœ2 1 Ê alnnnb q 1 n . Thus 3, ln n 1 nœ2 _ q ! aln nb diverges by Comparison Test. Thus, if _ q _ and p Ÿ 1, nœ2 n q the series ! alnnpncbr diverges. n œ1 63. Converges by Exercise 61 with q œ 3 and p œ 4. 64. Diverges by Exercise 62 with q œ 12 and p œ 12 . 65. Converges by Exercise 61 with q œ 1000 and p œ 1.001. 66. Diverges by Exercise 62 with q œ 15 and p œ 0.99. 67. Converges by Exercise 61 with q œ 3 and p œ 1.1. 68. Diverges by Exercise 62 with q œ 12 and p œ 12 . 69. Example CAS commands: Maple: a := n -> 1./n^3/sin(n)^2; s := k -> sum( a(n), n=1..k ); # (a)] limit( s(k), k=infinity ); pts := [seq( [k,s(k)], k=1..100 )]: # (b) plot( pts, style=point, title="#69(b) (Section 10.4)" ); pts := [seq( [k,s(k)], k=1..200 )]: # (c) plot( pts, style=point, title="#69(c) (Section 10.4)" ); pts := [seq( [k,s(k)], k=1..400 )]: # (d) plot( pts, style=point, title="#69(d) (Section 10.4)" ); evalf( 355/113 ); Mathematica: Clear[a, n, s, k, p] a[n_]:= 1 / ( n3 Sin[n]2 ) s[k_]= Sum[ a[n], {n, 1, k}] points[p_]:= Table[{k, N[s[k]]}, {k, 1, p}] points[100] ListPlot[points[100]] points[200] ListPlot[points[200] points[400] ListPlot[points[400], PlotRange Ä All] To investigate what is happening around k = 355, you could do the following. N[355/113] N[1 355/113] Sin[355]//N a[355]//N N[s[354]] Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.5 The Ratio and Root Tests 597 N[s[355]] N[s[356]] _ _ 70. (a) Let S œ ! n12 , which is a convergent p-series. By Example 5 in Section 10.2, ! nan 1 1b converges to 1. By Theorem 8, nœ1 nœ1 _ _ _ _ nœ1 _ nœ1 nœ1 nœ1 S œ ! n12 œ ! nan 1 1b ! n12 ! nan 1 1b _ _ nœ1 nœ1 œ ! nan 1 1b ! Š n12 nan 1 1b ‹ also converges. _ _ nœ1 nœ1 (b) Since ! nan 1 1b converges to 1 (from Example 5 in Section 10.2), S œ 1 ! Š n12 nan 1 1b ‹ œ 1 ! n2 an1 1b nœ1 _ (c) _ Ä 0 faster than the terms of ! n12 . The new series is comparible to ! n13 , so it will converge faster because its terms nœ1 1000 nœ1 1000 (d) The series 1 ! n2 an1 1b gives a better approximation. Using Mathematica, 1 ! n2 an1 1b œ 1.644933568, while nœ1 1000000 ! nœ1 nœ1 1 1 n2 œ 1.644933067. Note that 6 œ 1.644934067. The error is 4.99 ‚ 107 compared with 1 ‚ 106 . 2 10.5 THE RATIO AND ROOT TESTS 1. 2. 3. 2n n! 0 for all n 2nb" 1; lim Œ n2n" ! œ nÄ_ a n nÄ_ n! b1b b 2 an 1; lim Œ 3nnbb21 œ n2 3n 0 for all n nÄ_ b1bc1b! b2 b b anb1b2 _ n œ1 _ nÄ_ aan a n n 3n aa nÄ_ 3 3 ‰ n3 ‰ ! n n 2 converges ˆ1‰ 1 lim ˆ n3 œ lim ˆ 3n n †3 † n 2 6 œ lim 3 œ 3 1 Ê 3 1; lim Œ nbnc1 1b!1 œ nÄ_ an 1 b ! 0 for all n an 1 b2 _ 2 †2 n! ! 2 converges ˆ 2 ‰ lim Š an" n! b†n! † 2n ‹ œ lim n " œ 0 1 Ê b nÄ_ nÄ_ 2 n œ1 "b 3n 4n 1 n 2n n lim Š na†nan21b2b! † aann 1b! ‹ œ lim Š n2 4n 4 ‹ œ lim Š 2n 4 ‹ 2 3 nÄ_ 2 nÄ_ nÄ_ 1 b! œ lim ˆ 6n 2 4 ‰ œ _ 1 Ê ! aann diverges 1 b2 nÄ_ 4. n œ1 2nb1 n†3n 1 0 for all n _ 2an1b1 b an1b 1 1; lim n1 2†3n1 nÄ_ n 1 a n†3 nb1 lim Š an21b†3†n2 1 †3 † n2†3n1 ‹ œ lim ˆ 3n2n 3 ‰ œ lim ˆ 23 ‰ œ 23 1 œ n 1 nÄ_ nÄ_ nÄ_ nb1 Ê ! n2†3nc1 converges n œ1 5. n4 4n 0 for all n b1b4 an 1; lim Œ 4nnb4 1 œ nÄ_ n 4 4 lim Š an4n †14b † n44 ‹ œ lim Š n 4n 4n6n4 4n 1 ‹ n nÄ_ 4 3 2 nÄ_ _ 4 œ lim ˆ 14 1n 2n3 2 n13 4n1 4 ‰ œ 14 1 Ê ! n4n converges nÄ_ 6. n œ1 3nb2 ln n 0 for all n 3anb1bb2 2; lim Œ ln3nnbb21 œ nÄ_ a ln n _ nb2 3 lim Š ln3an †31b † 3lnnbn2 ‹ œ lim Š ln 3anlnn1b ‹ œ lim Š 1n ‹ œ lim ˆ 3n n 3 ‰ b nÄ_ nÄ_ nÄ_ nb1 nÄ_ nb2 œ lim ˆ 31 ‰ œ 3 1 Ê ! 3ln n diverges nÄ_ 7. n œ2 n 2 an 2 b ! 0 for all n nx32n an 1; lim nÄ_ b 1b2aan b 1b b 2b! b 1bx32an 1b an n2 an 2b! nx32n œ 2 3ban 2b! lim Š an an1ba1nb † n2 annx3 2b! ‹ œ lim Š n 9n5n3 9n7n2 3 ‹ †nx32n †32 2n 3 nÄ_ nÄ_ _ 7 ! n an 2n2b! converges ˆ 6n 15 ‰ ˆ6‰ 1 œ lim Š 3n27n2 15n 18n ‹ œ lim 54n 18 œ lim 54 œ 9 1 Ê n x3 2 nÄ_ nÄ_ nÄ_ 2 n œ1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 2 598 8. Chapter 10 Infinite Sequences and Series an n †5 n a2n 3b lnan 1b 0 for all n a a nÄ_ 1b†a2n 3b lim Š 5an † lnlnaann 21bb ‹ œ na2n 5b œ b lim Œ 2 n 1 1; b 1b†5n 1 3b lnaan 1b n†5n 3b lnan 1b a2n a2n 3b lnan 1b n 1 b† 5 † 5 lim Š a2na ‹ 5b lnan 2b † n †5 n n œ 1b nÄ_ 1 a b 15 25 ‰ lim Š 10n 2n2 25n † lim Š n b1 1 ‹ ‹ † lim Š lnlnann 21b ‹ œ lim ˆ 20n 4n 5 5n 2 nÄ_ nÄ_ nÄ_ _ nÄ_ nÄ_ nb2 n †5 ! ‰ ˆ n2‰ ˆ 1‰ œ lim ˆ 20 4 † lim n 1 œ 5 † lim 1 œ 5 † 1 œ 5 1 Ê a2n 3b lnan 1b diverges nÄ_ 9. 7 a2n 5bn nÄ_ nÄ_ n 10. a3n4 bn 0 for all n 3‰ 11. ˆ 4n 3n 5 n n 1 _ nÄ_ n œ1 _ 4 n 1; lim É œ a3n bn nÄ_ nÄ_ n œ1 n n 3‰ ˆ 4n 2; lim É œ 3n 5 n 1 n nÄ_ diverges 8 2n 3 1n ‰ n 1; lim É ˆ n 3‰ lim ˆ 43 ‰ œ 43 1 Ê ! ˆ 4n diverges 3n 5 nÄ_ 1; lim Ê’lnê2 1n ‰“ Ê ! ’lnê2 1n ‰“ _ 3‰ lim ˆ 4n 3n 5 œ nÄ_ 0 for all n n 4 ‰ lim ˆ 3n œ 0 1 Ê ! a3n4 bn converges n n 1 _ n È lim Š 2n 7 5 ‹ œ 0 1 Ê ! a2n 7 5bn converges nÄ_ 0 for all n 12. ’lnê2 1n ‰“ n œ2 7 n 1; lim É œ a2n 5bn 0 for all n n nÄ_ n œ1 1 1 În œ lim ’lnê2 1n ‰“ nÄ_ œ lnae2 b œ 2 1 n œ1 13. ˆ 0 for all n nÄ_ n 14. ’sinŠ È1n ‹“ 15. ˆ1 1n ‰ 16. n1"bn n2 8 2n œ 3 1n ‰ lim Œ ˆ nÄ_ n È 8 0 for all n n2 17. converges by the Ratio Test: nÄ_ n œ1 _ n nÄ_ nÄ_ n n2 lim ˆ1 1n ‰ œ e1 1 Ê ! ˆ1 1n ‰ converges 1; lim Éˆ1 1n ‰ œ n " 2; lim É n1bn œ n lim sinŠ È1n ‹ œ sina0b œ 0 1 Ê ! ’sinŠ È1n ‹“ converges nÄ_ n 8 n œ1 _ n 0 for all n 1 n n 1; lim Ê ’sinŠ È1n ‹“ œ 0 for all n _ œ 9 1 Ê ! ˆ3 1 ‰2n converges 3 1 ‰ 2 nÄ_ n œ1 n È nÄ_ ” lim anb1 œ n lim n Ä _ an Ä_ _ n È 1 ! 1"bn converges lim Š n È n n‹ œ 0 1 Ê n lim Š n1În 1 1 ‹ œ nÄ_ È (n b 1) 2 2nb1 • È n 2 n œ2 È È n (n 1) 2 ˆ1 n" ‰ 2 ˆ #" ‰ œ #" 1 œ n lim † 2È2 œ n lim Ä _ #nb1 Ä_ n ” #n • Š (nenbb1)1 ‹ 2 18. converges by the Ratio Test: lim anb1 œ n lim n Ä _ an Ä_ anb1 19. diverges by the Ratio Test: n lim œ n lim Ä _ an Ä_ anb1 20. diverges by the Ratio Test: n lim œ n lim Ä _ an Ä_ # Š nen ‹ Š (nenbb1)! 1 ‹ ˆ en!n ‰ b 1)! ‹ Š (n 10nb1 ˆ 10n!n ‰ "! 21. converges by the Ratio Test: lim anb1 œ n lim n Ä _ an Ä_ "! 2 n (n ")! e n" œ n lim † n! œ n lim e œ_ Ä _ enb1 Ä_ n (n ")! 10 n œ n lim † n! œ n lim œ_ Ä _ 10nb1 Ä _ 10 Š (n10bn1)1 ‹ Š n10n ‹ # (n 1) lim ˆ1 n" ‰ ˆ "e ‰ œ "e 1 œ n lim † ne2 œ n Ä _ Ä _ enb1 n "! (n ") " ˆ1 "n ‰"! ˆ 1"0 ‰ œ 10 œ n lim † 10 1 n"! œ n lim Ä _ 10n 1 Ä_ n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.5 The Ratio and Root Tests n 599 n ˆ nn 2 ‰ œ lim ˆ1 n2 ‰ œ e# Á 0 22. diverges; n lim a œ n lim Ä_ n Ä_ nÄ_ (1) ˆ 45 ‰ c2 (1)n d Ÿ ˆ 45 ‰ (3) which is the nth term of a convergent 23. converges by the Direct Comparison Test: 2(1.25) n œ n n n geometric series 24. converges; a geometric series with krk œ ¸ 23 ¸ 1 n n ˆ1 3n ‰ œ lim ˆ1 n3 ‰ œ e$ ¸ 0.05 Á 0 25. diverges; n lim a œ n lim Ä_ n Ä_ nÄ_ " ‰n ˆ1 3n 26. diverges; n lim a œ n lim œ n lim 1 Ä_ n Ä_ Ä_ Š "3 ‹ n n "Î$ œe 27. converges by the Direct Comparison Test: lnn$n nn$ œ n"# for n ¸ 0.72 Á 0 2, the nth term of a convergent p-series. n 1În n a(ln n) b n (ln n) n È É 28. converges by the nth-Root Test: n lim an œ n lim n 1În nn œ n lim Ä_ Ä_ Ä_ an b ln n œ n lim œ n lim Ä_ n Ä_ Š "n ‹ 1 œ01 29. diverges by the Direct Comparison Test: "n n"# œ n n# 1 "# ˆ "n ‰ for n 2 or by the Limit Comparison Test (part 1) with "n . n 1În n n n ˆ n" n"# ‰ œ lim ˆˆ n" n"# ‰ ‰ È É 30. converges by the nth-Root Test: n lim an œ n lim Ä_ Ä_ nÄ_ 31. diverges by the Direct Comparison Test: lnnn n" for n ˆ " n"# ‰ œ 0 1 œ n lim Ä_ n 3 (n 1) ln (n 1) anb1 32. converges by the Ratio Test: n lim œ n lim † n ln2 (n) œ "# 1 #nb1 Ä _ an Ä_ n (n 2)(n 3) anb1 n! 33. converges by the Ratio Test: n lim œ n lim † (n 1)(n (n 1)! 2) œ 0 1 Ä _ an Ä_ $ (n 1) anb1 34. converges by the Ratio Test: n lim œ n lim † ne$ œ "e 1 Ä _ an Ä _ en 1 n (n 4)! anb1 n4 35. converges by the Ratio Test: n lim œ n lim † 3! n! 3 œ n lim œ "3 1 Ä _ an Ä _ 3! (n 1)! 3nb1 (n 3)! Ä _ 3(n 1) n nb1 (n 1)2 (n 2)! anb1 2 2‰ ˆ n n 1 ‰ ˆ 32 ‰ ˆ nn 36. converges by the Ratio Test: n lim œ n lim † n2n3(n n! 3nb1 (n 1)! 1 œ 3 1 1)! œ n lim Ä _ an Ä_ Ä_ n (n 1)! 1)! anb1 n" 37. converges by the Ratio Test: n lim œ n lim † (2n n! œ n lim œ01 Ä _ an Ä _ (2n 3)! Ä _ (2n 3)(2n 2) (n 1)! anb1 " ˆ n ‰ œ lim 38. converges by the Ratio Test: n lim œ n lim † n œ n lim Ä _ an Ä _ (n 1)nb1 n! Ä _ n1 n Ä _ ˆ n b " ‰n n n n œ n lim Ä_ ˆ " n 1 "n ‰ œ "e 1 n n È n " n n È 39. converges by the Root Test: n lim an œ n lim œ n lim œ n lim œ01 Ä_ Ä _ É (ln n)n Ä _ ln n Ä _ ln n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 600 Chapter 10 Infinite Sequences and Series n n È n n n È 40. converges by the Root Test: n lim an œ n lim œ n lim Ä_ Ä _ É (ln n)nÎ2 Ä_ È ln n Ä_ Ä_ n n lim È n Èln n œ 0 1 lim n œ n È n œ 1‹ Šn lim Ä_ n! ln n ln n n " " 41. converges by the Direct Comparison Test: n(n 2)! œ n(n 1)(n 2) n(n 1)(n 2) œ (n 1)(n #) n# which is the nth-term of a convergent p-series $ n n 1 3 an 1 3 n ˆ 3 ‰ œ #3 1 42. diverges by the Ratio Test: n lim œ n lim † n32n œ n lim Ä _ an Ä _ (n 1)$ 2n 1 Ä _ (n 1)3 # an1bx‘2 2 an 1 b anb1 n 2n 1 43. converges by the Ratio Test: n lim œ n lim † a2nbx œ n lim œ n lim œ 14 1 Ä _ an Ä _ 2(n 1)‘x ‘2 Ä _ (2n 2)(2n 1) Ä _ 4n2 6n 2 2 nx anb1 44. converges by the Ratio Test: n lim œ n lim Ä _ an Ä_ n n n n a2n 5bˆ2nb1 3‰ † a2n 33ba22n 3b œ n lim ’ 2n 5 † 2†6 4†2 3†3 6 “ 3nb1 2 Ä _ 2n 3 3†6n 9†3n 2†2n 6 œ n lim ’ 2n 5 “ † n lim ’ 2†6 4†2 3†3 6 “ œ 1 † 23 œ 23 1 Ä _ 2n 3 Ä _ 3 †6 n 9 † 3 n 2 † 2 n 6 n n n anb1 45. converges by the Ratio Test: n lim œ n lim Ä _ an Ä_ anb1 46. converges by the Ratio Test: n lim œ n lim Ä _ an Ä_ ˆ 1 b nsin n ‰ an œ01 an "n Š 1 b tan n ‹ an an approaches 1 1# while the denominator tends to _ anb1 47. diverges by the Ratio Test: n lim œ n lim Ä _ an Ä_ " tan œ n lim n Ä_ "n œ 0 since the numerator c1‰ ˆ 3n 3n 1 2n b 5 an œ n lim œ #3 1 an Ä _ 2n 5 2 ‰ 48. diverges; an1 œ n n 1 an Ê an1 œ ˆ n n 1 ‰ ˆ n n 1 an1 ‰ Ê an1 œ ˆ n n 1 ‰ ˆ n n 1 ‰ ˆ nn 1 an2 a " n n 1 n 2 3 " Ê an1 œ ˆ n 1 ‰ ˆ n ‰ ˆ n 1 ‰ â ˆ # ‰ a" Ê an1 œ n 1 Ê an1 œ n 1 , which is a constant times the general term of the diverging harmonic series anb1 49. converges by the Ratio Test: n lim œ n lim Ä _ an Ä_ 50. converges by the Ratio Test: lim anb1 œ n lim n Ä _ an Ä_ anb1 51. converges by the Ratio Test: n lim œ n lim Ä _ an Ä_ Š 2n ‹ an an 2 œ n lim œ01 Ä_ n Èn n Œ # an an œ n lim Ä_ Š 1 bnln n ‹ an an n n È n œ "# 1 "ln n " œ n lim œ n lim œ01 n Ä_ Ä_ n 52. nnln10n 0 and a" œ #" Ê an 0; ln n 10 for n e"! Ê n ln n n 10 Ê nnln10n 1 Ê an1 œ nnln10n an an ; thus an1 an " # Ê n lim a Á 0, so the series diverges by the nth-Term Test Ä_ n 3 3 6 " %! " 2 " 2 " 2 " É É É É 53. diverges by the nth-Term Test: a" œ "3 , a# œ É 3 , a$ œ Ê 3 œ 3 , a% œ ËÊ 3 œ 3 ,á , % n! " n! " n " É an œ É a œ 1 because šÉ 3 Ê n lim 3 › is a subsequence of š 3 › whose limit is 1 by Table 8.1 Ä_ n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.5 The Ratio and Root Tests # $ # ' % ' #% 54. converges by the Direct Comparison Test: a" œ "# , a# œ ˆ "# ‰ , a$ œ Šˆ "# ‰ ‹ œ ˆ "# ‰ , a% œ Šˆ "# ‰ ‹ œ ˆ "# ‰ , á n! n Ê an œ ˆ "# ‰ ˆ "# ‰ which is the nth-term of a convergent geometric series 2 anb1 55. converges by the Ratio Test: n lim œ n lim Ä _ an Ä_ nb1 n" œ n lim œ #" 1 Ä _ 2n 1 (n 1)! (n 1)! † 2(2n)! lim 2(n 1)(n 1) n n! n! œ (2n 2)! n Ä _ (2n #)(2n 1) (3n 3)! 1)! (n 2)! anb1 56. diverges by the Ratio Test: n lim œ n lim † n! (n (3n)! Ä _ an Ä _ (n 1)! (n 2)! (n 3)! (3n 3)(3 2)(3n 1) 2 ‰ ˆ 3n 1 ‰ œ n lim œ n lim 3 ˆ 3n n# n 3 œ 3 † 3 † 3 œ 27 1 Ä _ (n 1)(n 2)(n 3) Ä_ n n! n (n!) n È 57. diverges by the Root Test: n lim an ´ n lim œ n lim œ_1 Ä_ Ä _ É an n b # Ä _ n# n! n (n!) n (n!) ˆ " ‰ ˆ 2n ‰ ˆ 3n ‰ â ˆ n n 1 ‰ ˆ nn ‰ É 58. converges by the Root Test: n lim œ n lim œ n lim œ n lim É an n b n Ä_ Ä_ Ä _ nn Ä_ n nn# n n " Ÿ n lim œ01 Ä_ n n " n n n È 59. converges by the Root Test: n lim an œ n lim œ n lim œ n lim œ01 Ä_ Ä _ É 2n# Ä _ #n Ä _ #n ln 2 n n n n n È 60. diverges by the Root Test: n lim an œ n lim œ n lim œ_1 Ä_ Ä _ É a#n b # Ä_ 4 n 1†3† â †(2n 1)(2n 1) anb1 4 2 n! 2n " 61. converges by the Ratio Test: n lim œ n lim † 1 †3 † â œ "4 1 4nb1 2nb1 (n 1)! †(2n 1) œ n lim Ä _ an Ä_ Ä _ (4†#)(n 1) n n (2n 1) 1†2†3†4 â (2n 1)(2n) (2n)! 62. converges by the Ratio Test: an œ (2†14†3ââ#n) a3n 1b œ (2†4 â 2n)# a3n 1b œ a2n n!b# a3n 1b # a3 1 b (2n ")(2n 2) a3 1b (2n 2)! Ê n lim † a2 n!b(2n)! œ n lim 2# (n 1)# a3n 1 1b Ä _ c2nb1 (n 1)!d# a3nb1 1b Ä_ n n n cn a1 3 b œ n lim œ 1 † 3" œ 3" 1 Š 4n 6n 2 ‹ Ä _ 4n# 8n 4 a3 3cn b # anb1 " ˆ n ‰ œ 1p œ 1 Ê no conclusion 63. Ratio: n lim œ n lim † n œ n lim Ä _ an Ä _ (n 1)p 1 Ä_ n1 p p " " n " n È É Root: n lim an œ n lim n n‰p œ (1)p œ 1 Ê no conclusion np œ n lim Ä_ Ä_ Ä _ ˆÈ p ˆ"‰ p anb1 ln n n" " n 64. Ratio: n lim œ n lim † (ln1n) œ ’n lim œ Šn lim “ œ ”n lim ‹ Ä _ an Ä _ (ln (n 1))p Ä _ ln (n 1) Ä _ ˆ n b" 1 ‰ • Ä_ n p œ (1)p œ 1 Ê no conclusion " n n È Root: n lim an œ n lim É (ln n)p œ Ä_ Ä_ " p lim (ln n)1În ‹ ŠnÄ_ ln (ln n) Ê n lim ln f(n) œ n lim œ n lim n Ä_ Ä_ Ä_ ˆ n ln" n ‰ 1 n È œ n lim eln fÐnÑ œ e! œ 1; therefore n lim an œ Ä_ Ä_ n) ; let f(n) œ (ln n)1În , then ln f(n) œ ln (ln n " œ n lim œ 0 Ê n lim (ln n)1În Ä _ n ln n Ä_ " p lim (ln n)1În ‹ ŠnÄ_ _ œ (1)" p œ 1 Ê no conclusion (n ") 2 65. an Ÿ 2nn for every n and the series ! #nn converges by the Ratio Test since n lim † n œ "# 1 Ä _ 2nb1 _ n nœ1 Ê ! an converges by the Direct Comparison Test nœ1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. p 601 602 Chapter 10 Infinite Sequences and Series 2 2anb1b n2 n2 b2nb1 1; lim n2bn21 ! œ nÄ_ 66. 2n! 0 for all n a nÄ_ n! _ 2nb1 †4 ‰ ˆ 2†4 1ln 4 ‰ lim Š a2n1b†n! † 2n!n2 ‹ œ lim Š 2n1 ‹ œ lim ˆ n2 1 œ lim b n nÄ_ n nÄ_ nÄ_ n2 œ _ 1 Ê ! 2n! diverges n œ1 10.6 ALTERNATING SERIES, ABSOLUTE AND CONDITIONAL CONVERGENCE 1. converges by the Alternating Convergence Test since: un œ È1n 0 for all n Ê Èn11 Ÿ È1n Ê un1 Ÿ un ; lim un œ nÄ_ 1; n 1Ê n1 _ _ nœ1 nœ1 n Ê Èn 1 Èn lim 1 œ 0. nÄ_ Èn " 2. converges absolutely Ê converges by the Alternating Convergence Test since ! kan k œ ! n$Î# which is a convergent p-series 3. converges Ê converges by Alternating Series Test since: un œ n31 n 0 for all n Ê an 1b3n1 n 3n Ê an11b3nb1 Ÿ n13n Ê un1 Ÿ un ; lim un œ lim un œ nÄ_ ln n Ê aln an 1bb2 2; n 3n 2Ên1 n aln nb2 Ê aln an11bb2 Ÿ aln1nb2 Ê aln an41bb2 Ÿ aln4nb2 Ê un1 Ÿ un ; lim 4 2 œ 0. nÄ_ aln nb 5. converges Ê converges by Alternating Series Test since: un œ n2 n 1 0 for all n Ê n3 2n2 2n Ê n2n1 n Ê 3n1 lim 1 n œ 0. nÄ_ n 3 nÄ_ 4. converges Ê converges by Alternating Series Test since: un œ aln4nb2 0 for all n Ê ln an 1b 1Ên1 1; n n3 n2 n 1 Ê nan2 2n 2b n 1 Ê un1 Ÿ un ; an 1 b 2 1 lim un œ nÄ_ 1 Ê 2n2 2n 1; n n3 n2 n 1 Ê nŠan 1b2 1‹ n2 n 1 an2 1ban 1b lim 2 n œ 0. nÄ_ n 1 6. diverges Ê diverges by nth Term Test for Divergence since: 2 lim n2 5 œ 1 Ê nÄ_ n 4 7. diverges Ê diverges by nth Term Test for Divergence since: lim 2n2 œ _ Ê n nÄ_ 5 lim a1bn1 nn2 4 œ does not exist 2 nÄ_ lim a1bn1 2n2 œ does not exist n nÄ_ _ _ nœ1 nœ1 n 8. converges absolutely Ê converges by the Absolute Convergence Test since ! kan k œ ! an10 1bx , which converges by the Ratio Test, since lim anabn 1 œ nÄ_ lim n 10 2 œ 0 1 nÄ_ _ n n n n ‰ ˆ n ‰ Á 0 Ê ! (1)n1 ˆ 10 9. diverges by the nth-Term Test since for n 10 Ê 10 1 Ê n lim diverges Ä _ 10 nœ1 10. converges by the Alternating Series Test because f(x) œ ln x is an increasing function of x Ê ln"x is decreasing Ê un un1 for n 1; also un 0 for n " 1 and n lim œ0 Ä _ ln n 11. converges by the Alternating Series Test since f(x) œ lnxx Ê f w (x) œ 1 x#ln x 0 when x e Ê f(x) is decreasing Ê un un1 ; also un 0 for n ln n 1 and n lim u œ n lim œ n lim Ä_ n Ä_ n Ä_ Š "n ‹ 1 œ0 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.6 Alternating Series, Absolute and Conditional Convergence 603 12. converges by the Alternating Series Test since f(x) œ ln a1 x" b Ê f w (x) œ x(x" 1) 0 for x 0 Ê f(x) is decreasing Ê un un1 ; also un 0 for n ˆ1 n" ‰‹ œ ln 1 œ 0 1 and n lim u œ n lim ln ˆ1 "n ‰ œ ln Šn lim Ä_ n Ä_ Ä_ 13. converges by the Alternating Series Test since f(x) œ Ê un unb1 ; also un 0 for n Èx " x1 1 and n lim u œ n lim Ä_ n Ä_ È 3 n1 14. diverges by the nth-Term Test since n lim œ n lim Ä _ Èn 1 Ä_ _ _ nœ1 nœ1 1 x 2È x Ê f w (x) œ 2Èx (x 1)# 0 Ê f(x) is decreasing Èn " n1 œ 0 3É1 "n 1 Š È"n ‹ œ3Á0 n " ‰ 15. converges absolutely since ! kan k œ ! ˆ 10 a convergent geometric series nb1 " ‰ 16. converges absolutely by the Direct Comparison Test since ¹ (1) n (0.1) ¹ œ (10)" n n ˆ 10 which is the nth term n n of a convergent geometric series _ _ nœ1 nœ1 " " 17. converges conditionally since È"n Èn" 1 0 and n lim œ 0 Ê convergence; but ! kan k œ ! n"Î# Ä _ Èn is a divergent p-series " 18. converges conditionally since 1 "Èn 1 È"n 1 0 and n lim œ 0 Ê convergence; but Ä _ 1 Èn _ _ nœ1 nœ1 ! kan k œ ! " is a divergent series since 1"Èn 1 Èn _ _ nœ1 nœ1 _ " " and ! n"Î# is a divergent p-series #È n nœ1 19. converges absolutely since ! kan k œ ! n$n1 and n$n1 n"# which is the nth-term of a converging p-series n! 20. diverges by the nth-Term Test since n lim œ_ Ä _ #n _ " " 21. converges conditionally since n " 3 (n 1) œ 0 Ê convergence; but ! kan k 3 0 and n lim Ä _ n 3 nœ1 _ œ! nœ1 " " n 3 diverges because n 3 _ " ! " is a divergent series 4n and n nœ1 _ 22. converges absolutely because the series ! ¸ sinn# n ¸ converges by the Direct Comparison Test since ¸ sinn# n ¸ Ÿ n"# nœ1 3n 23. diverges by the nth-Term Test since n lim œ1Á0 Ä _ 5n nb1 nb1 24. converges absolutely by the Direct Comparison Test since ¹ (n2)5n ¹ œ n25n 2 ˆ 25 ‰ which is the nth term n of a convergent geometric series 25. converges conditionally since f(x) œ x"# x" Ê f w (x) œ ˆ x2$ x"# ‰ 0 Ê f(x) is decreasing and hence un unb1 0 for n _ _ nœ1 nœ1 _ _ nœ1 nœ1 ˆ " n" ‰ œ 0 Ê convergence; but ! kan k œ ! 1n#n 1 and n lim Ä _ n# œ ! n"# ! n" is the sum of a convergent and divergent series, and hence diverges Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 604 Chapter 10 Infinite Sequences and Series 26. diverges by the nth-Term Test since n lim a œ n lim 101În œ 1 Á 0 Ä_ n Ä_ 27. converges absolutely by the Ratio Test: n lim Š uunbn 1 ‹ œ n lim Ä_ Ä_ ” (n")# ˆ 23 ‰ n n# ˆ 23 ‰ n 1 •œ 3 1 2 1d 28. converges conditionally since f(x) œ x ln" x Ê f w (x) œ cln(x(x) ln x)# 0 Ê f(x) is decreasing " 2 and n lim œ 0 Ê convergence; but by the Integral Test, Ä _ n ln n Ê un unb1 0 for n '2_ x dxln x œ lim '2b Šln x‹ dx œ lim cln (ln x)d b2 œ lim cln (ln b) ln (ln 2)d œ _ " x bÄ_ _ _ nœ1 nœ1 bÄ_ bÄ_ Ê ! kan k œ ! n ln" n diverges _ # 29. converges absolutely by the Integral Test since '1 atan" xb ˆ 1 " x# ‰ dx œ lim ’ atan # xb “ " bÄ_ œ lim ’atan bÄ_ " # " bb atan œ (x ln x)# œ n lim Ä_ Š "n ‹ 1 Š n" ‹ _ _ nœ1 nœ1 ! kan k œ ! 1 # # 1# 1b “ œ "# ’ˆ 1# ‰ ˆ 14 ‰ “ œ 332 # 30. converges conditionally since f(x) œ x lnlnx x Ê f w (x) œ 1 Š lnxx ‹ ln x Š lnxx ‹ b œ (x1lnln x)x # 0 Ê un Š "x ‹ (x ln x) (ln x) Š1 x" ‹ (x ln x)# ln n un1 0 when n e and n lim Ä _ n ln n œ 0 Ê convergence; but n ln n n Ê n"ln n n" Ê nlnlnn n n" so that ln n n ln n diverges by the Direct Comparison Test n 31. diverges by the nth-Term Test since n lim œ1Á0 Ä _ n1 _ _ nœ1 nœ1 n 32. converges absolutely since ! kan k œ ! ˆ 5" ‰ is a convergent geometric series nb1 ("00) n! 33. converges absolutely by the Ratio Test: n lim † (100) lim "00 œ 0 1 Š uunbn 1 ‹ œ n lim n œ Ä_ Ä _ (n1)! n Ä _ n1 _ _ nœ1 nœ1 " " " 34. converges absolutely by the Direct Comparison Test since ! kan k œ ! n# 2n 1 and n# 2n 1 n# which is the nth-term of a convergent p-series _ _ nœ1 nœ1 _ " 35. converges absolutely since ! kan k œ ! ¹ (nÈ1)n ¹ œ ! n$Î# is a convergent p-series n nœ1 _ _ nœ1 nœ1 36. converges conditionally since ! cosnn1 œ ! (n1) is the convergent alternating harmonic series, but _ _ nœ1 nœ1 ! kan k œ ! n " n diverges 1) n È kan k œ n lim 37. converges absolutely by the Root Test: n lim Š (n(2n) n ‹ Ä_ Ä_ n 1 În n" œ n lim œ "# 1 Ä _ #n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.6 Alternating Series, Absolute and Conditional Convergence # 605 # a(n 1)!b (n 1) 38. converges absolutely by the Ratio Test: n lim † (2n)! œ n lim œ "4 1 ¹ anb1 ¹ œ n lim Ä _ an Ä _ ((2n 2)!) (n!)# Ä _ (2n 2)(2n 1) (2n)! (n ")(n 2)â(2n) 39. diverges by the nth-Term Test since n lim kan k œ n lim œ n lim 2n n Ä_ Ä _ 2n n! n Ä_ (n 1)(n 2)â(n (n 1)) ˆ n # 1 ‰ œ n lim n lim #nc1 Ä_ Ä_ n 1 œ_Á0 (n 1)! (n 1)! 3 40. converges absolutely by the Ratio Test: n lim ¹ anabn 1 ¹ œ n lim (2n 3)! Ä_ Ä_ nb1 1)! † (2n n! n! 3n # (n 1) 3 œ n lim œ 43 1 Ä _ (2n 2)(2n 3) Èn 1 Èn Èn 1 Èn † Èn 1 Èn œ Èn "1 Èn and š Èn 1" Èn › is a 1 41. converges conditionally since _ (") decreasing sequence of positive terms which converges to 0 Ê ! Èn converges; but 1 Èn n nœ1 _ _ nœ1 nœ1 " Èn 1 Èn ! kan k œ ! diverges by the Limit Comparison Test (part 1) with È"n ; a divergent p-series: " lim nÄ_ Èn 1 Èn œ lim " Èn n Ä _ Èn Èn 1 Èn œ n lim Ä_ 1 œ "# É1 1n 1 È # 42. diverges by the nth-Term Test since n lim ŠÈn# n n‹ œ n lim ŠÈn# n n‹ † Š Èn# nn ‹ Ä_ Ä_ n n n n œ n lim œ n lim Ä _ È n# n n Ä_ " É1 "n 1 œ #" Á 0 43. diverges by the nth-Term Test since n lim ŠÉn Èn Èn‹ œ n lim ŠÉn Èn Èn‹ Ä_ Ä_ – œ n lim Ä_ Èn É n Èn Èn œ n lim Ä_ É n Èn Èn É n È n È n — " œ #" Á 0 " É 1 Èn 1 44. converges conditionally since š Èn "Èn 1 › is a decreasing sequence of positive terms converging to 0 _ (") Ê ! Èn Èn 1 converges; but n lim Ä_ n nœ1 _ " Èn 1 ‹ Èn " œ n lim œ n lim œ #" Ä _ È n È n 1 Ä _ 1 É 1 " Š È"n ‹ n Š Èn _ so that ! Èn "Èn 1 diverges by the Limit Comparison Test with ! È"n which is a divergent p-series nœ1 nœ1 n n 2 45. converges absolutely by the Direct Comparison Test since sech (n) œ en 2ecn œ e2n2e 1 2e e2n œ en which is the nth term of a convergent geometric series _ _ nœ1 nœ1 46. converges absolutely by the Limit Comparison Test (part 1): ! kan k œ ! en 2ecn Apply the Limit Comparison Test with e1n , the n-th term of a convergent geometric series: 2 cn 2e 2 e ce lim œ n lim œ n lim œ2 n Ä _ Œ 1n Ä _ en ecn Ä _ 1 ec2n n n e _ nb1 ) 1 1 1 1 47. 14 16 18 10 12 14 Þ Þ Þ œ ! 2(" an 1b ; converges by Alternating Series Test since: un œ 2an 1b 0 for all n nœ1 n2 n 1 Ê 2an 2b 2an 1b Ê 2aan 11b 1b Ÿ 2an11b Ê un1 Ÿ un ; lim un œ nÄ_ lim 1 œ 0. nÄ_ 2an1b Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 1; 606 Chapter 10 Infinite Sequences and Series _ _ _ nœ1 nœ1 nœ1 1 1 1 1 1 48. 1 14 19 16 25 36 49 64 Þ Þ Þ œ ! an ; converges by the Absolute Convergence Test since ! kan k œ ! n"# which is a convergent p-series 49. kerrork ¸(1)' ˆ "5 ‰¸ œ 0.2 50. kerrork ¸(1)' ˆ 10" & ‰¸ œ 0.00001 & "" 51. kerrork ¹(1)' (0.01) 5 ¹ œ 2 ‚ 10 52. kerrork k(1)% t% k œ t% 1 53. kerrork 0.001 Ê un1 0.001 Ê an 11b2 3 0.001 Ê an 1b2 3 1000 Ê n 1 È997 ¸ 30.5753 Ê n 54. kerrork 0.001 Ê un1 0.001 Ê an n 1b21 1 0.001 Ê an 1b2 1 1000an 1b Ê n ¸ 998.9999 Ê n 998È9982 4a998b 2 999 55. kerrork 0.001 Ê un1 0.001 Ê 1 È n 1‹ 3 0.001 Ê Šan 1b 3 ˆ an 1 b 3 È n 1 ‰ 2 È Ê ŠÈn 1‹ 3Èn 1 10 0 Ê Èn 1 œ 3 29 40 œ 2 Ê n œ 3 Ê n 3 1000 4 56. kerrork 0.001 Ê un1 0.001 Ê lnalnan1 3bb 0.001 Ê lnalnan 3bb 1000 Ê n 3 ee 1000 ¸ 5.297 ‚ 10323228467 which is the maximum arbitrary-precision number represented by Mathematica on the particular computer solving this problem.. ' " 57. (2n)! 105 ' Ê (2n)! 105 œ 200,000 Ê n ' 58. n!" 105 ' Ê 105 n! Ê n 59. (a) an 31 9 Ê 1 1 #"! 3!" 4!" 5!" 6!" 7!" 8!" ¸ 0.367881944 an1 fails since 3" #" _ _ nœ1 nœ1 5 Ê 1 #"! 4!" 6!" 8!" ¸ 0.54030 n _ n _ n n (b) Since ! kan k œ ! ˆ 3" ‰ ˆ #" ‰ ‘ œ ! ˆ 3" ‰ ! ˆ #" ‰ is the sum of two absolutely convergent nœ1 nœ1 series, we can rearrange the terms of the original series to find its sum: " ˆ "3 9" 27 á ‰ ˆ #" 4" 8" á ‰ œ ˆ "3 ‰ 1 ˆ "3 ‰ ˆ"‰ 1 #ˆ " ‰ œ #" 1 œ #" # " " 60. s#! œ 1 "# 3" 4" á 19 20 ¸ 0.6687714032 Ê s#! #" † #"1 ¸ 0.692580927 _ 61. The unused terms are ! (1)j 1 aj œ (1)n 1 aan 1 an 2 b (1)n 3 aan 3 an 4 b á jœn 1 œ (1)n 1 caan 1 an 2 b aan 3 an 4 b á d . Each grouped term is positive, so the remainder has the same sign as (1)n 1 , which is the sign of the first unused term. n n kœ1 kœ1 62. sn œ 1"†2 #"†3 3"†4 á n(n " 1) œ ! k(k " 1) œ ! ˆ k" k " 1 ‰ œ ˆ1 "# ‰ ˆ "# 3" ‰ ˆ 3" 4" ‰ ˆ 4" 5" ‰ á ˆ n" n " 1 ‰ which are the first 2n terms of the first series, hence the two series are the same. Yes, for n sn œ ! ˆ k" k" 1 ‰ œ ˆ1 #" ‰ ˆ #" 3" ‰ ˆ 3" 4" ‰ ˆ 4" 5" ‰ á ˆ n " 1 n" ‰ ˆ n" n " 1 ‰ œ 1 n " 1 kœ1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.6 Alternating Series, Absolute and Conditional Convergence 607 ˆ1 n " 1 ‰ œ 1 Ê both series converge to 1. The sum of the first 2n 1 terms of the first Ê n lim s œ n lim Ä_ n Ä_ ˆ1 n " 1 ‰ œ 1. series is ˆ1 n " 1 ‰ n " 1 œ 1. Their sum is n lim s œ n lim Ä_ n Ä_ _ _ _ _ nœ1 nœ1 nœ1 nœ1 63. Theorem 16 states that ! kan k converges Ê ! an converges. But this is equivalent to ! an diverges Ê ! kan k diverges _ _ nœ1 nœ1 64. ka" a# á an k Ÿ ka" k ka# k á kan k for all n; then ! kan k converges Ê ! an converges and these imply that _ _ nœ1 nœ1 º ! an º Ÿ ! kan k _ 65. (a) ! kan bn k converges by the Direct Comparison Test since kan bn k Ÿ kan k kbn k and hence nœ1 _ ! aan bn b converges absolutely nœ1 _ _ _ (b) ! kbn k converges Ê ! bn converges absolutely; since ! an converges absolutely and nœ1 _ nœ1 nœ1 _ _ ! bn converges absolutely, we have ! can (bn )d œ ! aan bn b converges absolutely by part (a) nœ1 _ _ _ nœ1 nœ1 _ nœ1 nœ1 nœ1 nœ1 (c) ! kan k converges Ê kkk ! kan k œ ! kkan k converges Ê ! kan converges absolutely _ _ _ nœ1 nœ1 nœ1 66. If an œ bn œ (1)n È"n , then ! (1)n È"n converges, but ! an bn œ ! n" diverges 67. s" œ "# , s# œ "# 1 œ "# , " " " " s$ œ "# 1 4" 6" 8" 10 1"# 14 16 18 #"0 2"# ¸ 0.5099, s% œ s$ 3" ¸ 0.1766, " " " " " " " s& œ s% #"4 #"6 #"8 30 3"# 34 36 38 40 42 44 ¸ 0.512, s' œ s& 5" ¸ 0.312, " " " " " " " " " " " s( œ s' 46 48 50 52 54 56 58 60 62 64 66 ¸ 0.51106 N" 1 68. (a) Since ! kan k converges, say to M, for % 0 there is an integer N" such that º ! kan k Mº #% nœ1 N" 1 N" 1 _ _ _ nœ1 nœ1 nœN" nœN" nœN" Í » ! kan k ! kan k ! kan k » #% Í » ! kan k» #% Í ! kan k #% . Also, ! an converges to L Í for % 0 there is an integer N# (which we can choose greater than or equal to N" ) such _ that ksN# Lk #% . Therefore, ! kan k #% and ksN# Lk #% . nœN" Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 608 Chapter 10 Infinite Sequences and Series _ k nœ1 nœ1 (b) The series ! kan k converges absolutely, say to M. Thus, there exists N" such that º ! kan k Mº % whenever k N" . Now all of the terms in the sequence ekbn kf appear in ekan kf. Sum together all of the N terms in ekbn kf, in order, until you include all of the terms ekan kf nœ" 1 , and let N# be the largest index in the N# N# _ nœ1 nœ1 nœ1 sum ! kbn k so obtained. Then º ! kbn k Mº % as well Ê ! kbn k converges to M. 10.7 POWER SERIES _ nb1 1. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ xxn ¹ 1 Ê kxk 1 Ê 1 x 1; when x œ 1 we have ! (1)n , a divergent Ä_ Ä_ nœ1 _ series; when x œ 1 we have ! 1, a divergent series nœ1 (a) the radius is 1; the interval of convergence is 1 x 1 (b) the interval of absolute convergence is 1 x 1 (c) there are no values for which the series converges conditionally nb1 2. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x(x5)5)n ¹ 1 Ê kx 5k 1 Ê 6 x 4; when x œ 6 we have Ä_ Ä_ _ _ nœ1 nœ1 ! (1)n , a divergent series; when x œ 4 we have ! 1, a divergent series (a) the radius is 1; the interval of convergence is 6 x 4 (b) the interval of absolute convergence is 6 x 4 (c) there are no values for which the series converges conditionally nb1 1) " " 3. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (4x (4x 1)n ¹ 1 Ê k4x 1k 1 Ê 1 4x 1 1 Ê # x 0; when x œ # we Ä_ Ä_ _ _ _ _ _ nœ1 nœ1 nœ1 nœ1 nœ1 have ! (1)n (1)n œ ! (1)2n œ ! 1n , a divergent series; when x œ 0 we have ! (1)n (1)n œ ! (1)n , a divergent series (a) the radius is "4 ; the interval of convergence is #" x 0 (b) the interval of absolute convergence is "# x 0 (c) there are no values for which the series converges conditionally nb1 4. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (3xn2)1 Ä_ Ä_ ˆ n ‰ 1 Ê k3x 2k 1 † (3x n 2)n ¹ 1 Ê k3x 2k n lim Ä _ n1 _ Ê 1 3x 2 1 Ê "3 x 1; when x œ 3" we have ! ("n ) which is the alternating harmonic series and is n nœ1 _ conditionally convergent; when x œ 1 we have ! "n , the divergent harmonic series nœ1 (a) the radius is "3 ; the interval of convergence is 3" Ÿ x 1 (b) the interval of absolute convergence is "3 x 1 (c) the series converges conditionally at x œ "3 nb1 kx 2 k 2) 5. n lim † (x 10 ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x 10 nb1 10 1 Ê kx 2k 10 Ê 10 x 2 10 2)n ¹ 1 Ê Ä_ Ä_ n _ _ nœ1 nœ1 Ê 8 x 12; when x œ 8 we have ! (")n , a divergent series; when x œ 12 we have ! 1, a divergent series (a) the radius is "0; the interval of convergence is 8 x 12 (b) the interval of absolute convergence is 8 x 12 (c) there are no values for which the series converges conditionally Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.7 Power Series nb1 609 6. n lim k2xk 1 Ê k2xk 1 Ê "# x "# ; when x œ "# we have ¹ uunbn 1 ¹ 1 Ê n lim ¹ (2x) (2x)n ¹ 1 Ê n lim Ä_ Ä_ Ä_ _ _ ! (")n , a divergent series; when x œ " we have ! 1, a divergent series # nœ1 nœ1 (a) the radius is "# ; the interval of convergence is "# x "# (b) the interval of absolute convergence is "# x "# (c) there are no values for which the series converges conditionally nb1 (n 2) (n 1)(n 2) 1)x 7. n lim 1 Ê kxk 1 ¹ uunbn 1 ¹ 1 Ê n lim ¹ (n (n 3) † nxn ¹ 1 Ê kxk n lim Ä_ Ä_ Ä _ (n 3)(n) _ Ê 1 x 1; when x œ 1 we have ! (")n n n # , a divergent series by the nth-term Test; when x œ " we nœ1 _ have ! n n # , a divergent series nœ1 (a) the radius is "; the interval of convergence is " x " (b) the interval of absolute convergence is " x " (c) there are no values for which the series converges conditionally nb1 8. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x n2)1 Ä_ Ä_ ˆ n ‰ 1 Ê kx 2k 1 † (x n 2)n ¹ 1 Ê kx 2k n lim Ä _ n1 _ Ê 1 x 2 1 Ê 3 x 1; when x œ 3 we have ! "n , a divergent series; when x œ " we have nœ1 _ ! (1) , a convergent series nœ1 n n (a) the radius is "; the interval of convergence is 3 x Ÿ " (b) the interval of absolute convergence is 3 x " (c) the series converges conditionally at x œ 1 nb1 x 9. n lim † ¹ uunbn 1 ¹ 1 Ê n lim ¹ Ä_ Ä _ (n 1)Èn 1 3nb1 nÈ n 3n xn ¹ 1 n n Ê k3xk Šn lim ‹ ŠÉn lim ‹1 Ä _ n1 Ä _ n1 _ ) Ê kx3k (1)(1) 1 Ê kxk 3 Ê 3 x 3; when x œ 3 we have ! (" , an absolutely convergent series; n$Î# n nœ1 _ 1 when x œ 3 we have ! n$Î# , a convergent p-series nœ1 (a) the radius is 3; the interval of convergence is 3 Ÿ x Ÿ 3 (b) the interval of absolute convergence is 3 Ÿ x Ÿ 3 (c) there are no values for which the series converges conditionally nb1 Èn n 10. n lim 1 Ê kx 1k 1 ¹ uunbn 1 ¹ 1 Ê n lim ¹ (xÈn1) 1 † (x 1)n ¹ 1 Ê kx 1k Én lim Ä_ Ä_ Ä _ n1 _ ) Ê 1 x 1 1 Ê 0 x 2; when x œ 0 we have ! (" , a conditionally convergent series; when x œ 2 n"Î# n nœ1 _ we have ! n"Î# , a divergent series 1 nœ1 (a) the radius is 1; the interval of convergence is 0 Ÿ x 2 (b) the interval of absolute convergence is 0 x 2 (c) the series converges conditionally at x œ 0 nb1 ˆ " ‰ 1 for all x 11. n lim † n! ¹ 1 Ê kxk n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ x Ä_ Ä _ (n 1)! xn Ä _ n1 (a) the radius is _; the series converges for all x Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 610 Chapter 10 Infinite Sequences and Series (b) the series converges absolutely for all x (c) there are no values for which the series converges conditionally nb1 nb1 ˆ " ‰ 1 for all x 12. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ 3 x † 3nn!xn ¹ 1 Ê 3 kxk n lim Ä_ Ä _ (n 1)! Ä _ n1 (a) the radius is _; the series converges for all x (b) the series converges absolutely for all x (c) there are no values for which the series converges conditionally nb1 2nb2 ˆ 4n ‰ œ 4x# 1 Ê x# 41 13. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ 4 n x 1 † 4n nx2n ¹ 1 Ê x# n lim Ä_ Ä_ Ä _ n1 _ n Ê 12 x 12 ; when x œ 12 we have ! 4n ˆ 12 ‰ 2n nœ1 _ ! nœ1 _ œ ! n1 , a divergent p-series; when x œ 12 we have nœ1 _ 4 ˆ 1 ‰2n œ ! n1 , a divergent p-series n 2 n nœ1 (a) the radius is 12 ; the interval of convergence is 12 x 12 (b) the interval of absolute convergence is 12 x 12 (c) there are no values for which the series converges conditionally nb1 14. n lim † n 3 ¹ 1 Ê lx 1l n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x 1) Š n ‹ œ 31 lx 1l 1 Ä_ Ä _ an 1b2 3nb1 (x 1)n Ä _ 3 an 1 b 2 2 _ n 2 _ Ê 2 x 4; when x œ 2 we have ! (n2 3)3n œ ! (n1) , an absolutely convergent series; when x œ 4 we have 2 n nœ1 _ n nœ1 _ n ! (3) ! 12 , an absolutely convergent series. 2 n œ nœ1 n 3 nœ1 n (a) the radius is 3; the interval of convergence is 2 Ÿ x Ÿ 4 (b) the interval of absolute convergence is 2 Ÿ x Ÿ 4 (c) there are no values for which the series converges conditionally nb1 x 15. n lim † ¹ uunbn 1 ¹ 1 Ê n lim ¹ Ä_ Ä _ È(n 1)# 3 È n# 3 ¹1 xn _ # n 3 Ê kxk Én lim " Ê kxk 1 Ä _ n# 2n 4 Ê 1 x 1; when x œ 1 we have ! È("# ) , a conditionally convergent series; when x œ 1 we have nœ1 _ " ! È# nœ1 n 3 n n 3 , a divergent series (a) the radius is 1; the interval of convergence is 1 Ÿ x 1 (b) the interval of absolute convergence is 1 x 1 (c) the series converges conditionally at x œ 1 n 1 x 16. n lim † ¹ uun n 1 ¹ 1 Ê n lim ¹ Ä_ Ä _ È(n 1)# 3 È n# 3 ¹1 xn # n 3 Ê kxk Én lim " Ê kxk 1 Ä _ n# 2n 4 _ _ nœ1 nœ1 ) Ê 1 x 1; when x œ 1 we have ! Èn"# 3 , a divergent series; when x œ 1 we have ! È(" , n# 3 a conditionally convergent series (a) the radius is 1; the interval of convergence is 1 x Ÿ 1 (b) the interval of absolute convergence is 1 x 1 (c) the series converges conditionally at x œ 1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. n Section 10.7 Power Series 3) 17. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (n 1)(x 5nb1 Ä_ Ä_ nb1 611 ˆ n n " ‰ 1 Ê kx 5 3k 1 † n(x 5 3)n ¹ 1 Ê kx 5 3k n lim Ä_ n _ _ Ê kx 3k 5 Ê 5 x 3 5 Ê 8 x 2; when x œ 8 we have ! n(5n5) œ ! (1)n n, a divergent n nœ1 _ nœ1 _ n series; when x œ 2 we have ! n5n œ ! n, a divergent series nœ1 5 nœ1 (a) the radius is 5; the interval of convergence is 8 x 2 (b) the interval of absolute convergence is 8 x 2 (c) there are no values for which the series converges conditionally nb1 # # (n 1) n 1 18. n lim † 4 annxn 1b ¹ 1 Ê kx4k n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (n 1)x ¹ a b ¹ 1 Ê kxk 4 Ä_ Ä _ 4nb1 an# 2n 2b Ä _ n an# 2n 2b n _ _ Ê 4 x 4; when x œ 4 we have ! n(n#1)1 , a conditionally convergent series; when x œ 4 we have ! n# n 1 , n nœ1 nœ1 a divergent series (a) the radius is 4; the interval of convergence is 4 Ÿ x 4 (b) the interval of absolute convergence is 4 x 4 (c) the series converges conditionally at x œ 4 19. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ Ä_ Ä_ Èn 1 xnb1 n † È3n xn ¹ 1 3nb1 ˆ n n 1 ‰ 1 Ê kx3k 1 Ê kxk 3 Ê kx3k Én lim Ä_ _ _ nœ1 nœ1 Ê 3 x 3; when x œ 3 we have ! (1)n Èn , a divergent series; when x œ 3 we have ! Èn, a divergent series (a) the radius is 3; the interval of convergence is 3 x 3 (b) the interval of absolute convergence is 3 x 3 (c) there are no values for which the series converges conditionally 20. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ Ä_ Ä_ nbÈ 1 n 1 (2x5)nb1 ¹1 n n (2x5)n È Ê k2x 5k n lim Š Ä_ nbÈ 1 n1 ‹1 n n È t lim È t Ä_ n 1 Ê k2x 5k 1 Ê 1 2x 5 1 Ê 3 x 2; when x œ 3 we have Ê k2x 5k Œ tlim Èn n _ Ä_ _ n n n ! (1) È È n, a divergent series since n lim n œ 1; when x œ 2 we have ! È n, a divergent series Ä_ nœ1 nœ1 (a) the radius is "# ; the interval of convergence is 3 x 2 (b) the interval of absolute convergence is 3 x 2 (c) there are no values for which the series converges conditionally _ _ _ nœ1 nœ1 21. First, rewrite the series as ! a2 (1)n bax 1bn1 œ ! 2ax 1bn1 ! (1)n ax 1bn1 . For the series nœ1 _ n ! 2ax 1bn1 : lim ¹ unb1 ¹ 1 Ê lim ¹ 2ax1nbc1 ¹ 1 Ê lx 1l lim 1 œ lx 1l 1 Ê 2 x 0; For the un nÄ_ n Ä _ 2 ax 1 b nÄ_ nœ1 _ nb1 n series ! (1)n ax 1bn1 : n lim 1 œ lx 1l 1 ¹ uunbn 1 ¹ 1 Ê n lim ¹ ( 1) ax 1b ¹ 1 Ê lx 1ln lim Ä_ Ä _ (1)n ax1bnc1 Ä_ nœ1 _ Ê 2 x 0; when x œ 2 we have ! a2 (1)n ba1bn1 , a divergent series; when x œ 0 we have nœ1 _ ! a2 (1)n b, a divergent series nœ1 (a) the radius is 1; the interval of convergence is 2 x 0 (b) the interval of absolute convergence is 2 x 0 (c) there are no values for which the series converges conditionally Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 612 Chapter 10 Infinite Sequences and Series ( 1) 22. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ Ä_ Ä_ 3 ax 2bnb1 9n † (1)n 32n3nax 2bn ¹ 1 Ê lx 2ln lim œ 9lx 2l 1 3 an 1 b Ä _ n1 nb1 2nb2 _ _ 19 17 ! (1) 3 ˆ 1 ‰ œ ! 1 , a divergent series; when x œ 19 we have Ê 17 9 x 9 ; when x œ 9 we have 3n 9 3n 9 n 2n nœ1 _ _ n nœ1 ! (1) 3 ˆ 1 ‰n œ ! (1) , a conditionally convergent series. nœ1 (a) (b) (c) n 2n 3n 9 n 3n nœ1 19 the radius is 19 ; the interval of convergence is 17 9 xŸ 9 19 the interval of absolute convergence is 17 9 x 9 the series converges conditionally at x œ 19 9 nb1 23. lim ¹ uunbn 1 ¹ 1 nÄ_ Ê n lim Ä_ » Š1 n " 1 ‹ xnb1 Š1 "n ‹ xn n " t lim Š1 t ‹ e Ä_ » 1 Ê kxk lim Š1 " ‹n 1 Ê kxk ˆ e ‰ 1 Ê kxk 1 n nÄ_ t _ n Ê 1 x 1; when x œ 1 we have ! (1)n ˆ1 "n ‰ , a divergent series by the nth-Term Test since nœ1 _ n n lim ˆ1 "n ‰ œ e Á 0; when x œ 1 we have ! ˆ1 n" ‰ , a divergent series nÄ_ nœ1 (a) the radius is "; the interval of convergence is 1 x 1 (b) the interval of absolute convergence is 1 x 1 (c) there are no values for which the series converges conditionally 24. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ ln (nxnln1)xn Ä_ Ä_ nb1 ˆ " ‰ n 1 ˆ n ‰ 1 Ê kxk 1 1 Ê kxk n lim ¹ 1 Ê kxk n lim Ä _ º ˆ"‰ º Ä _ n1 n _ Ê 1 x 1; when x œ 1 we have ! (1)n ln n, a divergent series by the nth-Term Test since n lim ln n Á 0; Ä_ nœ1 _ when x œ 1 we have ! ln n, a divergent series nœ1 (a) the radius is 1; the interval of convergence is 1 x 1 (b) the interval of absolute convergence is 1 x 1 (c) there are no values for which the series converges conditionally nb1 nb1 x 25. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (n 1) nn xn Ä_ Ä_ n ˆ1 n" ‰ ‹ Š lim (n 1)‹ 1 ¹ 1 Ê kxk Šn lim Ä_ nÄ_ Ê e kxk n lim (n 1) 1 Ê only x œ 0 satisfies this inequality Ä_ (a) the radius is 0; the series converges only for x œ 0 (b) the series converges absolutely only for x œ 0 (c) there are no values for which the series converges conditionally 26. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (n n!1)!(x(x4)4)n Ä_ Ä_ nb1 (n 1) 1 Ê only x œ 4 satisfies this inequality ¹ 1 Ê kx 4k n lim Ä_ (a) the radius is 0; the series converges only for x œ 4 (b) the series converges absolutely only for x œ 4 (c) there are no values for which the series converges conditionally nb1 ˆ n ‰ 1 Ê kx # 2k 1 Ê kx 2k 2 27. n lim † n2 ¹ 1 Ê kx # 2k n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x 2) Ä_ Ä _ (n 1) 2nb1 (x 2)n Ä _ n1 n _ _ nœ1 nœ1 ! (1) Ê 2 x 2 2 Ê 4 x 0; when x œ 4 we have ! " n , a divergent series; when x œ 0 we have n the alternating harmonic series which converges conditionally (a) the radius is 2; the interval of convergence is 4 x Ÿ 0 (b) the interval of absolute convergence is 4 x 0 (c) the series converges conditionally at x œ 0 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. nb1 , Section 10.7 Power Series nb1 613 nb1 (n 2)(x 1) ˆ n 2 ‰ 1 Ê 2 kx 1k 1 28. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ ((2)2)n (n 1)(x 1)n ¹ 1 Ê 2 kx 1k n lim Ä_ Ä_ Ä _ n1 _ Ê kx 1k "# Ê "# x 1 "# Ê "# x 3# ; when x œ "# we have ! (n 1) , a divergent series; when x œ 3# _ nœ1 we have ! (1) (n 1), a divergent series n n œ1 (a) the radius is "# ; the interval of convergence is "# x 3# (b) the interval of absolute convergence is "# x 3# (c) there are no values for which the series converges conditionally nb1 # # x n ln n 29. n lim † n(lnxnn) ¹ 1 Ê kxk Šn lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ ‹ Šn lim ‹ 1 Ä_ Ä _ (n 1) aln (n 1)b# Ä _ n1 Ä _ ln (n 1) ˆ"‰ # # n1 n Ê kxk (1) Œn lim 1 Ê kxk Šn lim ‹ 1 Ê kxk 1 Ê 1 x 1; when x œ 1 we have Ä _ ˆ " ‰ Ä_ n nb1 _ _ nœ1 nœ1 n " ! (1) # which converges absolutely; when x œ 1 we have ! n(ln n) n(ln n)# which converges (a) the radius is "; the interval of convergence is 1 Ÿ x Ÿ 1 (b) the interval of absolute convergence is 1 Ÿ x Ÿ 1 (c) there are no values for which the series converges conditionally nb1 ln (n) x n 30. n lim † n lnxn(n) ¹ 1 Ê kxk Šn lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ ‹ Šn lim ‹1 Ä_ Ä _ (n 1) ln (n 1) Ä _ n1 Ä _ ln (n 1) _ Ê kxk (1)(1) 1 Ê kxk 1 Ê 1 x 1; when x œ 1 we have ! (nln1)n , a convergent alternating series; n nœ2 _ when x œ 1 we have ! n ln" n which diverges by Exercise 38, Section 9.3 nœ2 (a) the radius is "; the interval of convergence is 1 Ÿ x 1 (b) the interval of absolute convergence is 1 x 1 (c) the series converges conditionally at x œ 1 2nb3 $Î# 5) n 31. n lim † (4x n 5)2n 1 ¹ 1 Ê (4x 5)# Šn lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (4x ‹ (n 1)$Î# Ä_ Ä_ Ä _ n1 _ $Î# 2nb1 Ê k4x 5k 1 Ê 1 4x 5 1 Ê 1 x 3# ; when x œ 1 we have ! (n1)$Î# nœ1 _ 1 Ê (4x 5)# 1 _ œ ! n" $Î# which is nœ1 2nb1 absolutely convergent; when x œ 3# we have ! ("n)$Î# , a convergent p-series nœ1 (a) the radius is "4 ; the interval of convergence is 1 Ÿ x Ÿ 3# (b) the interval of absolute convergence is 1 Ÿ x Ÿ 3# (c) there are no values for which the series converges conditionally nb2 32. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (3x2n1)4 Ä_ Ä_ ˆ 2n 2 ‰ 1 Ê k3x 1k 1 † (3x2n1)2nb1 ¹ 1 Ê k3x 1k n lim Ä _ 2n 4 _ nb1 1) Ê 1 3x 1 1 Ê 23 x 0; when x œ 23 we have ! (2n 1 , a conditionally convergent series; nœ1 _ when x œ 0 we have ! nœ1 _ (")nb1 ! " , a divergent series 2n 1 œ #n 1 nœ1 (a) the radius is "3 ; the interval of convergence is 23 Ÿ x 0 (b) the interval of absolute convergence is 23 x 0 (c) the series converges conditionally at x œ 23 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 614 Chapter 10 Infinite Sequences and Series nb1 x ˆ 1 ‰ 1 for all x 33. n lim † 2†4†6xân a2nb ¹ 1 Ê kxk n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ Ä_ Ä _ 2†4†6âa2nba2an 1bb Ä _ 2n 2 (a) the radius is _; the series converges for all x (b) the series converges absolutely for all x (c) there are no values for which the series converges conditionally 3 5 7 a2n 1ba2an 1b 1bx 34. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ † † â an 1b2 2nb1 Ä_ Ä_ nb2 a2n 3bn † 3†5†7âan2n2 1bxnb1 ¹ 1 Ê kxk n lim Š 2an 1b2 ‹ 1 Ê only Ä_ 2 2 n x œ 0 satisfies this inequality (a) the radius is 0; the series converges only for x œ 0 (b) the series converges absolutely only for x œ 0 (c) there are no values for which the series converges conditionally _ n an 1 b n n 35. For the series ! 121222â and 12 22 â n2 œ nan 1ba6 2n 1b so that we can â n2 x , recall 1 2 â n œ 2 nœ1 nan b 1b _ _ nb1 rewrite the series as ! Œ n n b 1 2 2n b 1 xn œ ! ˆ 2n 3 1 ‰xn ; then n lim † a2n3xn 1b ¹ 1 ¹ uunbn 1 ¹ 1 Ê n lim ¹ 3x Ä_ Ä _ a2 an 1 b 1 b a nœ1 Ê kxk ba b nœ1 6 _ lim ¹ a2n 1b ¹ 1 Ê kxk 1 Ê 1 x 1; when x œ 1 we have ! ˆ 2n 3 1 ‰a1bn , a conditionally n Ä _ a2n 3b nœ1 _ convergent series; when x œ 1 we have ! ˆ nœ1 3 ‰ 2n 1 , a divergent series. (a) the radius is 1; the interval of convergence is 1 Ÿ x 1 (b) the interval of absolute convergence is 1 x 1 (c) the series converges conditionally at x œ 1 _ 36. For the series ! ŠÈn 1 Èn‹ax 3bn , note that Èn 1 Èn œ nœ1 _ Èn 1 Èn Èn 1 Èn † Èn 1 Èn œ Èn 11 Èn so that we 1 nb1 n x3 can rewrite the series as ! Ènax13b Èn ; then n lim † ¹ uunbn 1 ¹ 1 Ê n lim ¹ a b Ä_ Ä _ Èn 2 Èn 1 Èn 1 Èn nœ1 ax 3 b n _ Èn 1 Èn ¹1 n b Ê lx 3ln lim 1 Ê lx 3l 1 Ê 2 x 4; when x œ 2 we have ! Èn a11 , a conditionally Èn Ä _ Èn 2 Èn 1 nœ1 _ convergent series; when x œ 4 we have ! Èn 11 Èn , a divergent series; nœ1 (a) the radius is 1; the interval of convergence is 2 Ÿ x 4 (b) the interval of absolute convergence is 2 x 4 (c) the series converges conditionally at x œ 2 nb1 lx l a n 1 bx x an 1 b 37. n lim † 3†6†n9xâxna3nb ¹ 1 Ê lxln lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ ¹ ¹ 1 Ê 3 1 Ê lxl 3 Ê R œ 3 Ä_ Ä _ 3†6†9âa3nba3an 1bb Ä _ 3 an 1 b 2 nb1 2 2 4 lx l 2 4 6 2n 2 n 1 x 38. n lim † aa22†5†4†8†6ââaa3n2nbb12 xbbn ¹ 1 Ê lxln lim ¹ a2n 2b ¹ 1 Ê 9 1 ¹ uunbn 1 ¹ 1 Ê n lim ¹ a † † âa ba a bbb Ä_ Ä _ a2†5†8âa3n 1ba3an 1b 1bb2 Ä _ a3n 2b2 Ê lxl 94 Ê R œ 94 2 nb1 2 lx l n 1 x an 1 b 39. n lim † 2 a2nbx ¹ 1 Ê lxln lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ a a bx b ¹ ¹ 1 Ê 8 1 Ê lxl 8 Ê R œ 8 Ä_ Ä _ 2nb1 a2an 1bbx anxb2 xn Ä _ 2a2n 2ba2n 1b n2 n n n Éˆ n n 1 ‰ xn 1 Ê lxl lim ˆ n n 1 ‰ 1 Ê lxle1 1 Ê lxl e Ê R œ e È 40. n lim un 1 Ê n lim Ä_ Ä_ nÄ_ n Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.7 Power Series nb1 615 nb1 41. n lim 3 1 Ê lxl 31 Ê 31 x 31 ; at x œ 31 we have ¹ uunbn 1 ¹ 1 Ê n lim ¹ 3 3n xxn ¹ 1 Ê lxl n lim Ä_ Ä_ Ä_ _ _ _ _ _ ! 3n ˆ 1 ‰n œ ! a1bn , which diverges; at x œ 1 we have ! 3n ˆ 1 ‰n œ ! 1 , which diverges. The series ! 3n xn 3 nœ0 3 nœ0 3 nœ0 nœ0 _ œ ! a3xbn is a convergent geometric series when 1 x 1 and the sum is 3 nœ0 3 nœ0 1 1 3x . nb1 e 4 42. n lim 1 1 Ê lex 4l 1 Ê 3 ex 5 Ê ln 3 x ln 5; ¹ uunbn 1 ¹ 1 Ê n lim ¹ a aex 4b bn ¹ 1 Ê lex 4l n lim Ä_ Ä_ Ä_ x _ n _ _ nœ0 nœ0 _ n at x œ ln 3 we have ! êln 3 4‰ œ ! a1bn , which diverges; at x œ ln 5 we have ! êln 5 4‰ œ ! 1, which nœ0 _ nœ0 diverges. The series ! aex 4bn is a convergent geometric series when ln 3 x ln 5 and the sum is 1 ae1x 4b œ 5 1 ex . nœ0 2nb2 43. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x 4n1)1 Ä_ Ä_ n † (x 4 1)2n ¹ 1 Ê (x 1)# lim 4 nÄ_ _ k1k 1 Ê (x 1)# 4 Ê kx 1k 2 _ _ Ê 2 x 1 2 Ê 1 x 3; at x œ 1 we have ! (42)n œ ! 44n œ ! 1, which diverges; at x œ 3 2n nœ0 _ _ 2n n nœ0 nœ0 _ n we have ! 24n œ ! 44n œ ! 1, a divergent series; the interval of convergence is 1 x 3; the series nœ0 _ ! nœ0 nœ0 nœ0 _ # n (x ")2n œ ! Šˆ x # 1 ‰ ‹ is a convergent geometric series when 1 x 3 and the sum is 4n nœ0 " œ " 1 Šxc # ‹ # " ’ 4 (x 4 ")# “ œ 4 x# 2x 1 œ 3 2x x# 4 4 2nb2 44. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x 9n1)1 Ä_ Ä_ n † (x 9 1)2n ¹ 1 Ê (x 1)# lim k1k 1 9 nÄ_ _ Ê (x 1)# 9 Ê kx 1k 3 _ Ê 3 x 1 3 Ê 4 x 2; when x œ 4 we have ! (93)n œ ! 1 which diverges; at x œ 2 we have nœ0 _ 2n nœ0 _ ! 32nn œ ! " which also diverges; the interval of convergence is 4 x 2; the series 9 nœ0 _ nœ0 _ n ! (x n1) œ ! Šˆ x1 ‰# ‹ is a convergent geometric series when 4 x 2 and the sum is 9 3 2n nœ0 nœ0 " 1Š # œ xb1 ‹ 3 " ’ 9 (x 1)# “ 9 œ 9 x# 9 2x 1 œ 8 2x9 x# 45. n lim ¹ uunbn 1 ¹ 1 Ê n lim Ä_ Ä_ º ˆÈx 2‰nb1 n † ˆÈx2 2‰n º 1 2nb1 Ê ¸È x 2 ¸ 2 Ê 2 È x 2 2 Ê 0 È x 4 _ _ Ê 0 x 16; when x œ 0 we have ! (1)n , a divergent series; when x œ 16 we have ! (1)n , a divergent nœ0 nœ0 _ Èx 2 n series; the interval of convergence is 0 x 16; the series ! Š # ‹ is a convergent geometric series when nœ0 0 x 16 and its sum is " 1Œ Èx c 2 œ # Œ 2c " Èx 2 # œ 4 2Èx nb1 46. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (ln(lnx)x)n ¹ 1 Ê kln xk 1 Ê 1 ln x 1 Ê e" x e; when x œ e" or e we Ä_ Ä_ _ _ _ nœ0 nœ0 nœ0 obtain the series ! 1n and ! (1)n which both diverge; the interval of convergence is e" x e; ! (ln x)n œ 1 "ln x " when e xe Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 616 Chapter 10 Infinite Sequences and Series # 47. n lim ¹ uunbn 1 ¹ 1 Ê n lim Š x 3 1 ‹ Ä_ Ä_ º n 1 # † ˆ x# 3 1 ‰ º 1 Ê ax 3 1b n lim k1k 1 Ê x 3 " 1 Ê x# 2 Ä_ n # _ Ê kxk È2 Ê È2 x È2 ; at x œ „ È2 we have ! (1)n which diverges; the interval of convergence is nœ0 _ # È2 x È2 ; the series ! Š x 3 1 ‹ n is a convergent geometric series when È2 x È2 and its sum is nœ0 " œ 3 c x"# c 1 œ # 3 x# # 1 Š x 3b 1 ‹ Š ‹ 3 # ax 1 b 48. n lim ¹ uun n 1 ¹ 1 Ê n lim ¹ 2n 1 Ä_ Ä_ n 1 † ax# 2 1bn ¹ 1 Ê kx# 1k 2 Ê È3 x È3 ; when x œ „ È3 we n _ _ n # have ! 1n , a divergent series; the interval of convergence is È3 x È3 ; the series ! Š x 2 1 ‹ is a nœ0 nœ0 convergent geometric series when È3 x È3 and its sum is nb1 49. n lim ¹ (x #n3) b1 Ä_ " " œ # 1 Šx 2 1‹ 2 œ 3 2 x# Šx# 1 ‹ # _ n † (x 2 3)n ¹ 1 Ê kx 3k 2 Ê 1 x 5; when x œ 1 we have ! (1)n which diverges; nœ1 _ when x œ 5 we have ! (1) which also diverges; the interval of convergence is 1 x 5; the sum of this n nœ1 " œ x2 1 . 3 1 Šxc # ‹ convergent geometric series is n If f(x) œ 1 #" (x 3) 4" (x 3)# á ˆ #" ‰ (x 3)n á n œ x 2 1 then f w (x) œ #" #" (x 3) á ˆ #" ‰ n(x 3)n1 á is convergent when 1 x 5, and diverges 2 2 when x œ 1 or 5. The sum for f w (x) is (x 1)# , the derivative of x 1 . 50. If f(x) œ 1 "# (x 3) 4" (x 3)# á ˆ #" ‰ (x 3)n á œ x 2 1 then ' f(x) dx n $ # œ x (x 4 3) (x 123) á ˆ "# ‰ _ n (x 3)n 1 á . n 1 _ At x œ 1 the series ! n21 diverges; at x œ 5 nœ1 2 the series ! (n1) 1 converges. Therefore the interval of convergence is 1 x Ÿ 5 and the sum is n nœ1 2 ln kx 1k (3 ln 4), since ' x 2 1 dx œ 2 ln kx 1k C, where C œ 3 ln 4 when x œ 3. # % ' ) "! 5x 7x 9x 11x 51. (a) Differentiate the series for sin x to get cos x œ 1 3x 3! 5! 7! 9! 11! á # % ' ) "! x œ 1 x#! x4! x6! x8! 10! á . The series converges for all values of x since lim nÄ_ 2nb2 n! " ¹ (2nx 2)! † a#x#8b ¹ œ x2 n lim Š ‹ œ 0 1 for all x. Ä _ a2n 1ba2n 2b $ $ & & ( ( * * "" "" $ & ( * "" 32x 128x 512x 2048x (b) sin 2x œ 2x 23!x 25!x 27!x 29!x 2 11!x á œ 2x 8x 3! 5! 7! 9! 11! á " "‰ $ ‰ # ˆ (c) 2 sin x cos x œ 2 (0 † 1) (0 † 0 1 † 1)x ˆ0 † " # 1 † 0 0 † 1 x 0 † 0 1 † # 0 † 0 1 † 3! x ˆ0 † 4!" 1 † 0 0 † #" 0 † 3!" 0 † 1‰ x% ˆ0 † 0 1 † 4!" 0 † 0 #" † 3!" 0 † 0 1 † 5!" ‰ x& $ & 16x ˆ0 † 6!" 1 † 0 0 † 4!" 0 † 3!" 0 † #" 0 † 5!" 0 † 1‰ x' á ‘ œ 2 ’x 4x 3! 5! á “ $ $ & & ( ( * * "" "" œ 2x 23!x 25!x 27!x 29!x 2 11!x á 52. (a) d 2x 3x# 4x$ 5x% x# x$ x% x x x x x ae b œ 1 2! 3! 4! 5! á œ 1 x #! 3! 4! á œ e ; thus the derivative of e is e itself (b) ' ex dx œ ex C œ x x# x3! x4! x5! á C, which is the general antiderivative of ex # $ % & (c) ex œ 1 x x#! x3! x4! x5! á ; ecx † ex œ 1 † 1 (1 † 1 1 † 1)x ˆ1 † #"! 1 † 1 #"! † 1‰ x# ˆ1 † 3!" 1 † #"! #"! † 1 3!" † 1‰ x$ ˆ1 † 4!" 1 † 3!" #"! † #"! 3!" † 1 4!" † 1‰ x% # $ % & ˆ1 † 5!" 1 † 4!" #"! † 3!" 3!" † #"! 4!" † 1 5!" † 1‰ x& á œ 1 0 0 0 0 0 á Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.8 Taylor and Maclaurin Series 617 17x 62x 53. (a) ln ksec xk C œ ' tan x dx œ ' Šx x3 2x 15 315 2835 á ‹ dx $ # % ' ) & ( * "! # % ' ) "! x 17x 31x x x 17x 31x œ x# 1x# 45 2520 14,175 á C; x œ 0 Ê C œ 0 Ê ln ksec xk œ x# 12 45 2520 14,175 á , converges when 1# x 1# $ & ( * % ' ) x) d 17x 62x 2x 17x 62x # (b) sec# x œ d(tan œ dx Šx x3 2x dx 15 315 2835 á ‹ œ 1 x 3 45 315 á , converges when 1# x 1# # % ' # % ' 61x x 5x 61x (c) sec# x œ (sec x)(sec x) œ Š1 x# 5x 24 720 á ‹ Š1 # 24 720 á ‹ 5 5 ‰ % 61 5 5 61 ‰ ' œ 1 ˆ "# "# ‰ x# ˆ 24 4" 24 x ˆ 720 48 48 720 x á % ' ) 1 1 62x œ 1 x# 2x3 17x 45 315 á , # x # 61x 54. (a) ln ksec x tan xk C œ ' sec x dx œ ' Š1 x2 5x 24 720 á ‹ dx # $ & ( * $ & ( * % ' x 61x 277x œ x x6 24 5040 72,576 á C; x œ 0 Ê C œ 0 Ê ln ksec x tan xk x 61x 277x œ x x6 24 5040 72,576 á , converges when 1# x 1# # % ' $ & ( x) d 61x 5x 61x 277x (b) sec x tan x œ d(sec œ dx Š1 x# 5x dx 24 720 á ‹ œ x 6 120 1008 á , converges when 1# x 1# # % ' $ & ( 61x x 2x 17x (c) (sec x)(tan x) œ Š1 x# 5x 24 720 á ‹ Šx 3 15 315 á ‹ $ & ( 2 5 ‰ & 17 " 5 61 ‰ ( 277x œ x ˆ "3 #" ‰ x$ ˆ 15 6" 24 x ˆ 315 15 72 720 x á œ x 5x6 61x 120 1008 á , 1# x 1# _ _ nœ0 nœk _ 55. (a) If f(x) œ ! an xn , then f ÐkÑ (x) œ ! n(n 1)(n 2)â(n (k 1)) an xnk and f ÐkÑ (0) œ k!ak Ê ÐkÑ ÐkÑ ak œ f k!(0) ; likewise if f(x) œ ! bn xn , then bk œ f k!(0) nœ0 Ê ak œ bk for every nonnegative integer k _ (b) If f(x) œ ! an xn œ 0 for all x, then f ÐkÑ (x) œ 0 for all x Ê from part (a) that ak œ 0 for every nonnegative integer k nœ0 10.8 TAYLOR AND MACLAURIN SERIES 1. f(x) œ e2x , f w (x) œ 2e2x , f ww (x) œ 4e2x , f www (x) œ 8e2x ; f(0) œ e2a0b œ ", f w (0) œ 2, f ww (0) œ 4, f www (0) œ 8 Ê P! (x) œ 1, P" (x) œ 1 2x, P# (x) œ 1 x 2x# , P$ (x) œ 1 x 2x# 43 x3 2. f(x) œ sin x, f w (x) œ cos x , f ww (x) œ sin x , f www (x) œ cos x; f(0) œ sin 0 œ 0, f w (0) œ 1, f ww (0) œ 0, f www (0) œ 1 Ê P! (x) œ 0, P" (x) œ x, P# (x) œ x, P$ (x) œ x 16 x3 3. f(x) œ ln x, f w (x) œ "x , f ww (x) œ x"# , f www (x) œ x2$ ; f(1) œ ln 1 œ 0, f w (1) œ 1, f ww (1) œ 1, f www (1) œ 2 Ê P! (x) œ 0, P" (x) œ (x 1), P# (x) œ (x 1) "# (x 1)# , P$ (x) œ (x 1) "# (x 1)# "3 (x 1)$ 4. f(x) œ ln (1 x), f w (x) œ 1 " x œ (1 x)" , f ww (x) œ (1 x)# , f www (x) œ 2(1 x)$ ; f(0) œ ln 1 œ 0, # # $ f w (0) œ 11 œ 1, f ww (0) œ (1)# œ 1, f www (0) œ 2(1)$ œ 2 Ê P! (x) œ 0, P" (x) œ x, P# (x) œ x x# , P$ (x) œ x x# x3 5. f(x) œ "x œ x" , f w (x) œ x# , f ww (x) œ 2x$ , f www (x) œ 6x% ; f(2) œ "# , f w (2) œ 4" , f ww (2) œ 4" , f www (x) œ 83 Ê P! (x) œ "# , P" (x) œ "# 4" (x 2), P# (x) œ #" 4" (x 2) 8" (x 2)# , " P$ (x) œ "# 4" (x 2) 8" (x 2)# 16 (x 2)$ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 618 Chapter 10 Infinite Sequences and Series 6. f(x) œ (x 2)" , f w (x) œ (x 2)# , f ww (x) œ 2(x 2)$ , f www (x) œ 6(x 2)% ; f(0) œ (2)" œ "# , f w (0) œ (2)# # œ 4" , f ww (0) œ 2(2)$ œ 4" , f www (0) œ 6(2)% œ 83 Ê P! (x) œ #" , P" (x) œ #" 4x , P# (x) œ #" 4x x8 , # $ x P$ (x) œ "# x4 x8 16 È2 È2 1 w ˆ1‰ 4 œ cos 4 œ # , # ,f È È È È È f ww ˆ 14 ‰ œ sin 14 œ #2 , f www ˆ 14 ‰ œ cos 14 œ #2 Ê P! œ #2 , P" (x) œ #2 #2 ˆx 14 ‰ , È È È È È È È # # $ P# (x) œ #2 #2 ˆx 14 ‰ 42 ˆx 14 ‰ , P$ (x) œ #2 #2 ˆx 14 ‰ 42 ˆx 14 ‰ 1#2 ˆx 14 ‰ 7. f(x) œ sin x, f w (x) œ cos x, f ww (x) œ sin x, f www (x) œ cos x; f ˆ 14 ‰ œ sin 14 œ 8. f(x) œ tan x, f w (x) œ sec2 x, f ww (x) œ 2sec2 x tan x, f www (x) œ 2sec4 x 4sec2 x tan2 x; f ˆ 14 ‰ œ tan 14 œ 1 , f w ˆ 14 ‰ œ sec2 ˆ 14 ‰ œ 2 , f ww ˆ 14 ‰ œ 2sec2 ˆ 14 ‰ tan ˆ 14 ‰ œ 4 , f www ˆ 14 ‰ œ 2sec4 ˆ 14 ‰ 4sec2 ˆ 14 ‰ tan2 ˆ 14 ‰ œ 16 Ê P! (x) œ 1 , 2 2 P" (x) œ 1 2 ˆx 14 ‰ , P# (x) œ 1 2 ˆx 14 ‰ 2 ˆx 14 ‰ , P$ (x) œ 1 2 ˆx 14 ‰ 2 ˆx 14 ‰ 83 ˆx 14 ‰ 3 9. f(x) œ Èx œ x"Î# , f w (x) œ ˆ "# ‰ x"Î# , f ww (x) œ ˆ 4" ‰ x$Î# , f www (x) œ ˆ 38 ‰ x&Î# ; f(4) œ È4 œ 2, " 3 f w (4) œ ˆ "# ‰ 4"Î# œ 4" , f ww (4) œ ˆ 4" ‰ 4$Î# œ 32 ,f www (4) œ ˆ 38 ‰ 4&Î# œ 256 Ê P! (x) œ 2, P" (x) œ 2 "4 (x 4), " " P# (x) œ 2 4" (x 4) 64 (x 4)# , P$ (x) œ 2 4" (x 4) 64 (x 4)# 51"# (x 4)$ 10. f(x) œ (1 x)"Î# , f w (x) œ "# (1 x)"Î# , f ww (x) œ 4" (1 x)$Î# , f www (x) œ 38 (1 x)&Î# ; f(0) œ (1)"Î# œ 1, f w (0) œ "# (1)"Î# œ "# , f ww (0) œ 4" (1)$Î# œ 4" , f www (0) œ 83 (1)&Î# œ 83 Ê P! (x) œ 1, 1 $ P" (x) œ 1 2" x, P# (x) œ 1 2" x 8" x# , P$ (x) œ 1 2" x 8" x# 16 x 11. f(x) œ ex , f w (x) œ ex , f ww (x) œ ex , f www (x) œ ex Ê á f ÐkÑ (x) œ a1bk ex ; f(0) œ ea0b œ ", f w (0) œ 1, _ f ww (0) œ 1, f www (0) œ 1, á ß f ÐkÑ (0) œ (1)k Ê ex œ 1 x 12 x# 16 x3 á œ ! (n!1) xn n nœ0 12. f(x) œ x ex , f w (x) œ x ex ex , f ww (x) œ x ex 2ex , f www (x) œ x ex 3ex Ê á f ÐkÑ (x) œ x ex k ex ; f(0) œ a0bea0b œ 0, _ f w (0) œ 1, f ww (0) œ 2, f www (0) œ 3, á ß f ÐkÑ (0) œ k Ê x x# 12 x3 á œ ! an 1 1b! xn nœ0 13. f(x) œ (1 x)" Ê f w (x) œ (1 x)# , f ww (x) œ 2(1 x)$ , f www (x) œ 3!(1 x)% Ê á f ÐkÑ (x) œ (1)k k!(1 x)k1 ; f(0) œ 1, f w (0) œ 1, f ww (0) œ 2, f www (0) œ 3!, á ß f ÐkÑ (0) œ (1)k k! _ _ nœ0 nœ0 Ê 1 x x# x$ á œ ! (x)n œ ! (1)n xn x 3 w ww $ www % 14. f(x) œ 21 Ê á f ÐkÑ (x) œ 3ak!b(1 x) k 1 ; f(0) œ 2, x Ê f (x) œ (1 x)# , f (x) œ 6(1 x) , f (x) œ 18(1 x) _ f w (0) œ 3, f ww (0) œ 6, f www (0) œ 18, á ß f ÐkÑ (0) œ 3ak!b Ê 2 3x 3x# 3x$ á œ 2 ! 3xn nœ1 _ n 2nb1 ) x 15. sin x œ ! (" (#n1)! nœ0 _ n 2nb1 ) x 16. sin x œ ! (" (#n1)! nœ0 _ 2nb1 Ê sin 3x œ ! ("(#) n(3x) 1)! n nœ0 _ Ê sin #x œ ! nœ0 _ 2n 1 (")n ˆ #x ‰ (#n1)! _ n 2nb1 2nb1 œ ! (")(#3n1)!x nœ0 _ n 2nb1 $ $ & & œ 3x 33!x 35!x á $ & œ ! #(2n"1)(2nx 1)! œ #x 2$x†3! 2&x†5! á nœ0 )x 7x 7x 7x 17. 7 cos (x) œ 7 cos x œ 7 ! (" (2n)! œ 7 #! 4! 6! á , since the cosine is an even function n 2n # % ' nœ0 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.8 Taylor and Maclaurin Series _ 1) x 18. cos x œ ! ((2n)! n 2n nœ0 _ (1x) Ê 5 cos 1x œ 5 ! (1)(#n)! œ 5 512!x 514!x 516!x á n # # 2n % % ' ' nœ0 cx œ "# ’Š1 x# x#! x3! x4! á ‹ Š1 x x#! x3! x4! á ‹“ œ 1 x#! x4! x6! á cx œ "# ’Š1 x x#! x3! x4! á ‹ Š1 x x#! x3! x4! á ‹“ œ x x3! x5! x6! á 19. cosh x œ e #e x _ 619 # $ % # $ % # % ' 2n x œ ! (2n)! nœ0 20. sinh x œ e #e x _ # $ % # $ % $ & ' 2n 1 œ ! (2nx 1)! nœ0 21. f(x) œ x% 2x$ 5x 4 Ê f w (x) œ 4x$ 6x# 5, f ww (x) œ 12x# 12x, f www (x) œ 24x 12, f Ð4Ñ (x) œ 24 Ê f ÐnÑ (x) œ 0 if n 5; f(0) œ 4, f w (0) œ 5, f ww (0) œ 0, f www (0) œ 12, f Ð4Ñ (0) œ 24, f ÐnÑ (0) œ 0 if n 5 24 % $ % $ Ê x% 2x$ 5x 4 œ 4 5x 12 3! x 4! x œ x 2x 5x 4 n 6 x 22. f(x) œ x x 1 Ê f w (x) œ a2x ; f ww (x) œ ax 2 1b3 ; f www (x) œ ax Ê f ÐnÑ (x) œ axa11bbnnbx 1 ; f(0) œ 0, f w (0) œ 0, f ww (0) œ 2, x 1 b2 1 b4 # # f www (0) œ 6, f ÐnÑ (0) œ a1bn nx if n _ 2 Ê x# x3 x4 x5 Þ Þ Þ œ ! a1bn xn nœ2 23. f(x) œ x$ 2x 4 Ê f w (x) œ 3x# 2, f ww (x) œ 6x, f www (x) œ 6 Ê f ÐnÑ (x) œ 0 if n 4; f(2) œ 8, f w (2) œ 10, 6 # $ f ww (2) œ 12, f www (2) œ 6, f ÐnÑ (2) œ 0 if n 4 Ê x$ 2x 4 œ 8 10(x 2) 12 2! (x 2) 3! (x 2) œ 8 10(x 2) 6(x 2)# (x 2)$ 24. f(x) œ 2x$ x# 3x 8 Ê f w (x) œ 6x# 2x 3, f ww (x) œ 12x 2, f www (x) œ 12 Ê f ÐnÑ (x) œ 0 if n f w (1) œ 11, f ww (1) œ 14, f www (1) œ 12, f ÐnÑ (1) œ 0 if n 4 Ê 2x$ x# 3x 8 12 # $ # $ œ 2 11(x 1) 14 2! (x 1) 3! (x 1) œ 2 11(x 1) 7(x 1) 2(x 1) 4; f(1) œ 2, 25. f(x) œ x% x# 1 Ê f w (x) œ 4x$ 2x, f ww (x) œ 12x# 2, f www (x) œ 24x, f Ð4Ñ (x) œ 24, f ÐnÑ (x) œ 0 if n 5; f(2) œ 21, f w (2) œ 36, f ww (2) œ 50, f www (2) œ 48, f Ð4Ñ (2) œ 24, f ÐnÑ (2) œ 0 if n 5 Ê x% x# 1 48 24 # $ % # $ % œ 21 36(x 2) 50 2! (x 2) 3! (x 2) 4! (x 2) œ 21 36(x 2) 25(x 2) 8(x 2) (x 2) 26. f(x) œ 3x& x% 2x$ x# 2 Ê f w (x) œ 15x% 4x$ 6x# 2x, f ww (x) œ 60x$ 12x# 12x 2, f www (x) œ 180x# 24x 12, f Ð4Ñ (x) œ 360x 24, f Ð5Ñ (x) œ 360, f ÐnÑ (x) œ 0 if n 6; f(1) œ 7, f w (1) œ 23, f ww (1) œ 82, f www (1) œ 216, f Ð4Ñ (1) œ 384, f Ð5Ñ (1) œ 360, f ÐnÑ (1) œ 0 if n 6 216 384 360 # $ % & Ê 3x& x% 2x$ x# 2 œ 7 23(x 1) 82 2! (x 1) 3! (x 1) 4! (x 1) 5! (x 1) œ 7 23(x 1) 41(x 1)# 36(x 1)$ 16(x 1)% 3(x 1)& 27. f(x) œ x# Ê f w (x) œ 2x$ , f ww (x) œ 3! x% , f www (x) œ 4! x& Ê f ÐnÑ (x) œ (1)n (n 1)! xn2 ; f(1) œ 1, f w (1) œ 2, f ww (1) œ 3!, f www (1) œ 4!, f ÐnÑ (1) œ (1)n (n 1)! Ê x"# _ œ 1 2(x 1) 3(x 1)# 4(x 1)$ á œ ! (1)n (n 1)(x 1)n nœ0 28. f(x) œ a1 1 xb3 Ê f w (x) œ 3(1 x)4 , f ww (x) œ 12(1 x)5 , f www (x) œ 60 (1 x)6 Ê f ÐnÑ (x) œ an 2 2b! (1 x)n3 ; fa0b œ 1, f w a0b œ 3, f ww a0b œ 12, f www a0b œ 60, á , f ÐnÑ a0b œ an 2 2b! Ê a1 1 xb3 œ 1 3x 6x# 10x3 á _ œ ! an 2ba2 n 1b xn nœ0 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 620 Chapter 10 Infinite Sequences and Series 29. f(x) œ ex Ê f w (x) œ ex , f ww (x) œ ex Ê f ÐnÑ (x) œ ex ; f(2) œ e# , f w (2) œ e# , á f ÐnÑ (2) œ e# # _ $ # Ê ex œ e# e# (x 2) e# (x 2)# e3! (x 2)$ á œ ! en! (x 2)n nœ0 30. f(x) œ 2x Ê f w (x) œ 2x ln 2, f ww (x) œ 2x (ln 2)# , f www (x) œ 2x (ln 2)3 Ê f ÐnÑ (x) œ 2x (ln 2)n ; f(1) œ 2, f w (1) œ 2 ln 2, f ww (1) œ 2(ln 2)# , f www (1) œ 2(ln 2)$ , á , f ÐnÑ (1) œ 2(ln 2)n # _ Ê 2x œ 2 (2 ln 2)(x 1) 2(ln# 2) (x 1)# 2(ln3!2) (x 1)3 á œ ! 2(ln 2)n!(x1) 3 n n nœ0 31. f(x) œ cosˆ2x 12 ‰, f w (x) œ 2 sinˆ2x 12 ‰, f ww (x) œ 4 cosˆ2x 12 ‰, f www (x) œ 8 sinˆ2x 12 ‰, f a4b axb œ 24 cosˆ2x 12 ‰ß f a5b axb œ 25 sinˆ2x 12 ‰ß . . ; fˆ 14 ‰ œ 1, f w ˆ 14 ‰ œ 0, f ww ˆ 14 ‰ œ 4, f www ˆ 14 ‰ œ 0, f a4b ˆ 14 ‰ œ 24 , 2 4 f a5b ˆ 14 ‰ œ 0, . . ., f Ð2nÑ ˆ 14 ‰ œ a1bn 22n Ê cosˆ2x 12 ‰ œ 1 2ˆx 14 ‰ 23 ˆx 14 ‰ . . . _ n 2n œ ! aa12nb b2x ˆx 14 ‰ 2n nœ0 7 Î2 32. f(x) œ Èx 1, f w (x) œ 12 ax 1b1Î2 , f ww (x) œ 14 ax 1b3Î2 , f www (x) œ 38 ax 1b5Î2 , f a4b (x) œ 15 , . . .; 16 ax 1b 1 1 3 15 1 1 1 5 f(0) œ 1, f w (0) œ , f ww (0) œ , f www (0) œ , f a4b (0) œ , . . . Ê Èx 1 œ 1 x x2 x3 x4 Þ Þ Þ 2 4 8 16 _ 2 8 16 128 n 33. The Maclaurin series generated by cos x is ! aa2n1bbx x2n which converges on a_, _b and the Maclaurin series generated nœ0 _ by 1 2 x is 2 ! xn which converges on a1, 1b. Thus the Maclaurin series generated by faxb œ cos x 1 2 x is given by nœ0 _ _ nœ0 nœ0 n ! a1b x2n 2 ! xn œ 1 2x 5 x2 Þ Þ Þ Þ which converges on the intersection of a_, _b and a1, 1b, so the a2nbx 2 interval of convergence is a1, 1b. _ n 34. The Maclaurin series generated by ex is ! xnx which converges on a_, _b. The Maclaurin series generated by nœ0 _ n faxb œ a1 x x be is given by a1 x x2 b ! xnx œ 1 12 x2 23 x3 Þ Þ Þ Þ which converges on a_, _bÞ 2 x nœ0 _ n a 1 b 2n1 35. The Maclaurin series generated by sin x is ! a2n which converges on a_, _b and the Maclaurin series 1 bx x nœ0 _ generated by lna1 xb is ! a1nb nœ1 nc1 xn which converges on a1, 1b. Thus the Maclaurin series genereated by _ _ n a 1 b a 1 b 2n1 faxb œ sin x † lna1 xb is given by Œ ! a2n Œ ! n 1 bx x nœ0 nœ1 nc1 xn œ x2 21 x3 61 x4 Þ Þ Þ Þ which converges on the intersection of a_, _b and a1, 1b, so the interval of convergence is a1, 1b. _ n a 1 b 2n1 36. The Maclaurin series generated by sin x is ! a2n which converges on a_, _b. The Maclaurin series 1 bx x nœ0 _ n 2 _ n _ n a 1 b a 1 b a 1 b 2n1 2n1 2n1 genereated by faxb œ x sin2 x is given by xŒ ! a2n œ xŒ ! a2n 1bx x Œ ! a2n 1bx x 1 bx x nœ0 nœ0 nœ0 2 7 œ x3 13 x5 45 x . . . which converges on a_, _bÞ _ ÐnÑ 37. If ex œ ! f n!(a) (x a)n and f(x) œ ex , we have f ÐnÑ (a) œ ea f or all n œ 0, 1, 2, 3, á nœ0 ! " # # Ê e œ ea ’ (x 0!a) (x 1!a) (x 2!a) á “ œ ea ’1 (x a) (x 2!a) á “ at x œ a x Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.9 Convergence of Taylor Series 621 38. f(x) œ ex Ê f ÐnÑ (x) œ ex for all n Ê f ÐnÑ (1) œ e for all n œ 0, 1, 2, á # $ Ê ex œ e e(x 1) #e! (x 1)# 3!e (x 1)$ á œ e ’1 (x 1) (x 2!1) (x 3!1) á “ ww www 39. f(x) œ f(a) f w (a)(x a) f #(a) (x a)# f 3!(a) (x a)$ á Ê f w (x) www Ð4Ñ œ f w (a) f ww (a)(x a) f 3!(a) 3(x a)# á Ê f ww (x) œ f ww (a) f www (a)(x a) f 4!(a) 4 † 3(x a)# á Ê f ÐnÑ (x) œ f ÐnÑ (a) f Ðn1Ñ (a)(x a) f Ðn 2Ñ (a) Ê f(a) œ f(a) 0, f w (a) œ f w (a) 0, á , f # Ðn Ñ (x a)# á (a) œ f ÐnÑ (a) 0 40. E(x) œ f(x) b! b" (x a) b# (x a)# b$ (x a)$ á bn (x a)n Ê 0 œ E(a) œ f(a) b! Ê b! œ f(a); from condition (b), lim xÄa f(x) f(a) b" (x a) b# (x a)# b$ (x a)$ á bn (x a)n œ0 (x a)n w # f (x) b" 2b# (x a) 3b$ (x a) á nbn (x a) Ê xlim n(x a)n 1 Äa n 1 œ0 f ww (x) 2b# 3! b$ (x a) á n(n ")bn (x a)n 2 Ê b" œ f (a) Ê xlim œ0 n(n 1)(x a)n 2 Äa www n 3 f (x) 3! b$ á n(n 1)(n 2)bn (x a) Ê b# œ "# f ww (a) Ê xlim œ0 n(n 1)(n #)(x a)n 3 Äa w f œ b$ œ 3!" f www (a) Ê xlim Äa ÐnÑ (x) n! bn œ0 n! Ê bn œ n!" f ÐnÑ (a); therefore, ww ÐnÑ g(x) œ f(a) f w (a)(x a) f 2!(a) (x a)# á f n!(a) (x a)n œ Pn (x) # 41. f(x) œ ln (cos x) Ê f w (x) œ tan x and f ww (x) œ sec# x; f(0) œ 0, f w (0) œ 0, f ww (0) œ 1 Ê L(x) œ 0 and Q(x) œ x2 42. f(x) œ esin x Ê f w (x) œ (cos x)esin x and f ww (x) œ ( sin x)esin x (cos x)# esin x ; f(0) œ 1, f w (0) œ 1, f ww (0) œ 1 # Ê L(x) œ 1 x and Q(x) œ 1 x x# 43. f(x) œ a1 x# b "Î# Ê f w (x) œ x a1 x# b $Î# and f ww (x) œ a1 x# b $Î# 3x# a1 x# b &Î# ; f(0) œ 1, f w (0) œ 0, # f ww (0) œ 1 Ê L(x) œ 1 and Q(x) œ 1 x# # 44. f(x) œ cosh x Ê f w (x) œ sinh x and f ww (x) œ cosh x; f(0) œ 1, f w (0) œ 0, f ww (0) œ 1 Ê L(x) œ 1 and Q(x) œ 1 x# 45. f(x) œ sin x Ê f w (x) œ cos x and f ww (x) œ sin x; f(0) œ 0, f w (0) œ 1, f ww (0) œ 0 Ê L(x) œ x and Q(x) œ x 46. f(x) œ tan x Ê f w (x) œ sec# x and f ww (x) œ 2 sec# x tan x; f(0) œ 0, f w (0) œ 1, f ww œ 0 Ê L(x) œ x and Q(x) œ x 10.9 CONVERGENCE OF TAYLOR SERIES _ # _ 5 x 5 x ! (1) 5 x 1. ex œ 1 x x#! á œ ! xn! Ê e5x œ 1 (5x) (#5x) ! á œ 1 5x #! 3! á œ n! # n # # $ $ nœ0 _ # nœ0 2. ex œ 1 x x#! á œ ! xn! Ê exÎ2 œ 1 ˆ #x ‰ n nœ0 & _ 3. sin x œ x x3! x5! á œ ! ((#1)nx1)! $ n 2n 1 nœ0 & _ 4. sin x œ x x3! x5! á œ ! ((#1)nx1)! $ n n n nœ0 n 2n 1 _ n n ˆ #x ‰# # $ á œ 1 x# 2x# #! 2x$ 3! á œ ! (21)n n!x #! nœ0 $ _ & x Ê 5 sin (x) œ 5 ’(x) (3!x) (5!x) á “ œ ! 5((1) #n1)! n 1 2n 1 nœ0 Ê sin 1#x œ 1#x _ n 2n 1 2n 1 ˆ 1#x ‰$ ˆ 1#x ‰& ˆ 1#x ‰( 1 x ! (1) 3! 5! 7! á œ 22n 1 (#n1)! nœ0 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 622 Chapter 10 Infinite Sequences and Series _ 1) x 5. cos x œ ! ((2n)! n 2n nœ0 _ Ê cos 5x2 œ ! nœ0 _ 6. 1bn x2n cos x œ ! a(2n)! Ê 5 x 625x 15625x œ ! (1)(2n)! œ 1 25x á #! 4! 6! n 2n 4n 4 8 12 nœ0 $Î# $ cos Š xÈ ‹ œ cos ŒŠ x# ‹ 2 nœ0 _ 2n (1)n 5x2 ‘ (2n)! "Î# $ "Î# a1bn ŒŠ x# ‹ _ œ! 2n _ nœ0 (1)n x3n 2n (2n)! œ! (#n)! n œ0 $ ' * œ 1 2x†2! 2#x†4! 2$x†6! á _ a1bnc1 ˆx2 ‰ n _ nc1 n 7. lna1 xb œ ! a1bn x Ê lna1 x2 b œ ! nœ1 nœ1 _ n 2nb1 1b x 8. tan1 x œ ! a2n 1 nœ0 9. _ a1bn ˆ3x4 ‰ 2n 1 Ê tan1 a3x4 b œ ! nœ0 _ n nc1 œ ! a1b n x œ x2 x2 x3 x4 . . . 2n 4 6 8 nœ1 2nb1 _ 2nb1 8nb4 n œ ! a 1 b 3 n x nœ0 _ _ _ nœ0 nœ0 nœ0 2187 28 20 œ 3x4 9x12 243 5 x 7 x ... 1 ! a1bn xn Ê 13 3 œ ! a1bn ˆ 3 x3 ‰n œ ! a1bn ˆ 3 ‰n x3n œ 1 3 x3 9 x6 27 x9 . . . 1x œ 4 4 4 16 64 1 4x _ _ _ n 10. 1 1 x œ ! xn Ê 2 1 x œ #" 1 1 " x œ #" ! ˆ #" x‰ œ ! ˆ #" ‰ # nœ0 _ nœ0 _ n _ n œ0 _ n 2nb1 1) x 12. sin x œ ! ((2n 1)! nœ0 _ 1) x 13. cos x œ ! ((2n)! n 2n nœ0 1 3 x œ #" 14 x 18 x2 16 x ... nœ0 xn ! xnb1 œ x x# x$ x% x& á #! n! œ n! 3! 4! 11. ex œ ! xn! Ê xex œ x Œ ! nœ0 n 1 n nœ0 _ _ n 2nb1 n 2nb3 1) x x x x $ Ê x# sin x œ x# Œ ! ((#1)nx1)! œ ! ((2n 1)! œ x 3! 5! 7! á nœ0 & ( * nœ0 _ x x x x x x x Ê x# 1 cos x œ x# 1 ! ((1) #n)! œ # 1 1 2 4! 6! 8! 10! á # # # n 2n # % ' ) "! nœ0 _ x x œ x4! x6! x8! 10! á œ ! ((1) #n)! % ' ) "! n 2n nœ2 _ n 2nb1 1) x 14. sin x œ ! ((2n 1)! nœ0 $ & _ $ n 2nb1 $ Ê sin x x x3! œ Œ ! ((#1)nx1)! x x3! nœ0 ( * "" $ & ( * _ "" 1) x x x œ Šx x3! x5! x7! x9! 11! á ‹ x x3! œ x5! x7! x9! 11! á œ ! ((2n 1)! n 2n 1 nœ2 _ 1) x 15. cos x œ ! ((2n)! n 2n nœ0 _ 1) x 16. cos x œ ! ((2n)! n 2n nœ0 _ _ n 2n 2nb1 (1x) Ê x cos 1x œ x ! (1)(#n)! œ ! (")(#1n)!x n 2n nœ0 nœ0 _ _ # 2n ax b x Ê x# cos ax# b œ x# ! (1)(#n)! œ ! ("(#) n)! n nœ0 _ n 4n 2 nœ0 % & ' ( ' "! "% œ x# x2! x4! x6! á % # # $ œ x 12!x 14!x 16!x á ' ) (2x) (2x) (2x) (2x) 17. cos# x œ "# cos#2x œ "# "# ! (1)(2n)! œ "# "# ’1 (2x) 2! 4! 6! 8! á “ n 2n nœ0 _ _ nœ1 nœ1 # n 2n n 2n 1 2n (2x)% (2x)' (2x)) ! (1) (2x) œ 1 ! (1) 2 x œ 1 (2x) 2†2! 2†4! 2†6! 2†8! á œ 1 2†(2n)! (2n)! # % ' # % ' (2x) (2x) (2x) (2x) (2x) 2x ‰ 18. sin# x œ ˆ 1cos œ "# "# cos 2x œ "# "# Š1 (2x) # #! 4! 6! á ‹ œ 2†2! 2†4! 2†6! á _ nb1 _ (2x) 2 x œ ! (1)#†(2n)! œ ! (1) (2n)! nœ1 2n n 2n 1 2n nœ1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.9 Convergence of Taylor Series _ _ nœ0 nœ0 # 19. 1x 2x œ x# ˆ 1"2x ‰ œ x# ! (2x)n œ ! 2n xn2 œ x# 2x$ 2# x% 2$ x& á _ nc1 _ nc1 n n 1 20. x ln (1 2x) œ x ! (1) n (2x) œ ! (1) n2 x nœ1 n nœ1 # $ $ % % & œ 2x# 2 #x 2 4x 2 5x á _ _ _ nœ0 nœ1 n œ0 d ˆ " ‰ " # ! nxn1 œ ! (n 1)xn 21. 1 " x œ ! xn œ 1 x x# x$ á Ê dx 1x œ (1x)# œ 1 2x 3x á œ _ # d ˆ " ‰ d " d # # ! n(n 1)xn2 22. a12xb$ œ dx # 1x œ dx Š (1x)# ‹ œ dx a1 2x 3x á b œ 2 6x 12x á œ nœ2 _ œ ! (n 2)(n 1)xn nœ0 3 5 7 23. tan1 x œ x 13 x3 15 x5 17 x7 Þ Þ Þ Ê x tan1 x2 œ xŠx2 13 ax2 b 15 ax2 b 17 ax2 b Þ Þ Þ ‹ _ n 4nc1 1b x œ x3 13 x7 15 x11 17 x15 Þ Þ Þ œ ! a2n 1 nœ1 3 5 7 b a2xb a2xb 24. sin x œ x x3! x5! x7! á Ê sin x † cos x œ "# sin 2x œ "# Š2x a2x 3! 5! 7! á ‹ 3 5 7 _ 2nb1 4x ! (1) 2 x œ x 43!x 165!x 647!x á œ x 23x 2x 15 315 á œ (#n1)! 3 5 7 3 5 7 n 2n nœ0 2 3 2 3 25. ex œ 1 x x2! x3! á and 1 1 x œ 1 x x2 x3 á Ê ex 1 1 x _ 4 ! ˆ 1 a1bn ‰xn œ Š1 x x2! x3! á ‹ a1 x x2 x3 á b œ 2 32 x2 56 x3 25 24 x á œ n! nœ0 3 5 7 2 4 6 26. sin x œ x x3! x5! x7! á and cos x œ 1 x2! x4! x6! á Ê cos x sin x 2 4 6 3 5 7 2 3 4 5 6 7 œ Š1 x2! x4! x6! á ‹ Šx x3! x5! x7! á ‹ œ 1 x x2! x3! x4! x5! x6! x7! á _ n 2nb1 1) x œ ! Š ((2n)! ((#1)nx1)! ‹ n 2n nœ0 2 3 4 27. lna1 xb œ x 12 x2 13 x3 14 x4 á Ê x3 lna1 x2 b œ x3 Šx2 12 ax2 b 13 ax2 b 14 ax2 b á ‹ _ nc1 1 9 œ 13 x3 16 x5 19 x7 12 x á œ ! a13nb x2n1 nœ1 28. lna1 xb œ x 12 x2 13 x3 14 x4 á and lna1 xb œ x 12 x2 13 x3 14 x4 á Ê lna1 xb lna1 xb _ œ ˆx 12 x2 13 x3 14 x4 á ‰ ˆx 12 x2 13 x3 14 x4 á ‰ œ 2x 23 x3 25 x5 á œ ! 2n 2 1 x2n1 nœ0 2 3 2 3 3 5 7 29. ex œ 1 x x2! x3! á and sin x œ x x3! x5! x7! á Ê ex † sin x 3 5 7 1 5 œ Š1 x x2! x3! á ‹Šx x3! x5! x7! á ‹ œ x x2 13 x3 30 x ÞÞÞÞ 30. lna1 xb œ x 12 x2 13 x3 14 x4 á and 1 " x œ 1 x x# x$ á Ê ln1a1xxb œ lna1 xb † 1 " x 7 4 œ ˆx 12 x2 13 x3 14 x4 á ‰a1 x x# x$ á b œ x 12 x2 56 x3 12 x ÞÞÞÞ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 623 624 Chapter 10 Infinite Sequences and Series 2 31. tan1 x œ x 13 x3 15 x5 17 x7 Þ Þ Þ Ê atan1 xb œ atan1 xbatan1 xb 44 8 6 œ ˆx 13 x3 15 x5 17 x7 Þ Þ Þ‰ˆx 13 x3 15 x5 17 x7 Þ Þ Þ‰ œ x2 23 x4 23 45 x 105 x Þ Þ Þ Þ 3 5 7 2 4 6 32. sin x œ x x3! x5! x7! á and cos x œ 1 x2! x4! x6! á Ê cos2 x † sin x œ cos x † cos x † sin x 3 5 7 b a2xb a2xb 7 3 61 5 1247 7 œ cos x † "# sin 2x œ "# Š1 x2! x4! x6! á ‹Š2x a2x 3! 5! 7! á ‹ œ x 6 x 120 x 5040 x Þ Þ Þ 2 3 5 4 6 7 2 3 33. sin x œ x x3! x5! x7! á and ex œ 1 x x2! x3! á 3 5 7 3 5 2 7 3 5 3 7 Ê esin x œ 1 Šx x3! x5! x7! á ‹ 12 Šx x3! x5! x7! á ‹ 16 Šx x3! x5! x7! á ‹ á œ 1 x 12 x2 18 x4 Þ Þ Þ Þ 34. sin x œ x x3! x5! x7! á and tan1 x œ x 13 x3 15 x5 17 x7 Þ Þ Þ Ê sinatan1 xb œ ˆx 13 x3 15 x5 17 x7 Þ Þ Þ‰ 3 5 7 3 5 7 1 ˆ 1 ˆ 16 ˆx 13 x3 15 x5 17 x7 Þ Þ Þ‰ 120 x 13 x3 15 x5 17 x7 Þ Þ Þ‰ 5040 x 13 x3 15 x5 17 x7 Þ Þ Þ‰ á 5 7 œ x 12 x3 38 x5 16 x ÞÞÞ 35. Since n œ 3, then f a4b axb œ sin x, lf a4b axbl Ÿ M on Ò0, 0.1Ó Ê lsin xl Ÿ 1 on Ò0, 0.1Ó Ê M œ 1. Then lR3 a0.1bl Ÿ 1 l0.14x 0l 4 œ 4.2 ‚ 106 Ê error Ÿ 4.2 ‚ 106 36. Since n œ 4, then f a5b axb œ ex , lf a5b axbl Ÿ M on Ò0, 0.5Ó Ê lex l Ÿ Èe on Ò0, 0.5Ó Ê M œ 2.7. Then lR4 a0.5bl Ÿ 2.7 l0.55x 0l œ 7.03 ‚ 104 Ê error Ÿ 7.03 ‚ 104 5 & 37. By the Alternating Series Estimation Theorem, the error is less than kx5!k Ê kxk& a5!b a5 ‚ 10% b Ê kxk& 600 ‚ 10% 5 Ê kxk È 6 ‚ 10# ¸ 0.56968 % # 38. If cos x œ 1 x# and kxk 0.5, then the error is less than ¹ (.5) 24 ¹ œ 0.0026, by Alternating Series Estimation Theorem; # since the next term in the series is positive, the approximation 1 x# is too small, by the Alternating Series Estimation Theorem c$ $ 39. If sin x œ x and kxk 10$ , then the error is less than a103! b ¸ 1.67 ‚ 1010 , by Alternating Series Estimation Theorem; $ The Alternating Series Estimation Theorem says R# (x) has the same sign as x3! . Moreover, x sin x Ê 0 sin x x œ R# (x) Ê x 0 Ê 10$ x 0. # $ # # x 40. È1 x œ 1 x# x8 16 á . By the Alternating Series Estimation Theorem the kerrork ¹ 8x ¹ (0.01) 8 œ 1.25 ‚ 10& c $ Ð0Þ1Ñ c $ $ 41. kR# (x)k œ ¹ e3!x ¹ 3 (0.1)$ 1.87 ‚ 104 , where c is between 0 and x 3! % 42. kR# (x)k œ ¹ e3!x ¹ (0.1) 3! œ 1.67 ‚ 10 , where c is between 0 and x # % ' # $ % & ' (2x) (2x) 2x ‰ 2x 2 x 2 x 43. sin# x œ ˆ 1 cos œ "# "# cos 2x œ "# "# Š1 (2x) # 2! 4! 6! á ‹ œ #! 4! 6! á # $ % & ' $ & ( (2x) (2x) (2x) d d 2 x 2 x Ê dx asin# xb œ dx Š 2x 2! 4! 6! á ‹ œ 2x 3! 5! 7! á Ê 2 sin x cos x $ & ( (2x) (2x) œ 2x (2x) 3! 5! 7! á œ sin 2x, which checks Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.9 Convergence of Taylor Series # % ' ) # $ % & ' 625 ( ) (2x) (2x) (2x) 2x 2 x 2 x 2 x 44. cos# x œ cos 2x sin# x œ Š1 (2x) #! 4! 6! 8! á ‹ Š #! 4! 6! 8! á ‹ # $ % & ' " % " 2 x 2 x 2 ' # ) œ 1 2x #! 4! 6! á œ 1 x 3 x 45 x 315 x á 45. A special case of Taylor's Theorem is f(b) œ f(a) f w (c)(b a), where c is between a and b Ê f(b) f(a) œ f w (c)(b a), the Mean Value Theorem. 46. If f(x) is twice differentiable and at x œ a there is a point of inflection, then f ww (a) œ 0. Therefore, L(x) œ Q(x) œ f(a) f w (a)(x a). ww 47. (a) f ww Ÿ 0, f w (a) œ 0 and x œ a interior to the interval I Ê f(x) f(a) œ f (c# # ) (x a)# Ÿ 0 throughout I Ê f(x) Ÿ f(a) throughout I Ê f has a local maximum at x œ a ww (b) similar reasoning gives f(x) f(a) œ f (c# # ) (x a)# 0 throughout I Ê f(x) f(a) throughout I Ê f has a local minimum at x œ a 48. f(x) œ (1 x)" Ê f w (x) œ (1 x)# Ê f ww (x) œ 2(1 x)$ Ê f Ð3Ñ (x) œ 6(1 x)% " 10 " ˆ 10 ‰ Ê f Ð4Ñ (x) œ 24(1 x)& ; therefore 1" x ¸ 1 x x# x$ . kxk 0.1 Ê 10 11 1x 9 Ê ¹ (1x)& ¹ 9 & % Ð4Ñ & % & Ð4Ñ ‰ Ê the error e$ Ÿ ¹ max f 4! (x) x ¹ (0.1)% ˆ 10 ‰ œ 0.00016935 0.00017, since ¹ f 4!(x) ¹ œ ¹ (1"x)& ¹ . Ê ¹ (1x x)& ¹ x% ˆ 10 9 9 49. (a) f(x) œ (1 x)k Ê f w (x) œ k(1 x)k1 Ê f ww (x) œ k(k 1)(1 x)k2 ; f(0) œ 1, f w (0) œ k, and f ww (0) œ k(k 1) Ê Q(x) œ 1 kx k(k # ") x# " " (b) kR# (x)k œ ¸ 3†3!2†" x$ ¸ 100 Ê kx$ k 100 Ê 0x " or 0 x .21544 100"Î$ 50. (a) Let P œ x 1 Ê kxk œ kP 1k .5 ‚ 10n since P approximates 1 accurate to n decimals. Then, P sin P œ (1 x) sin (1 x) œ (1 x) sin x œ 1 (x sin x) Ê k(P sin P) 1k $ 3n œ ksin x xk Ÿ kx3!k 0.125 .5 ‚ 103n Ê P sin P gives an approximation to 1 correct to 3n decimals. 3! ‚ 10 _ _ nœ0 nœk 51. If f(x) œ ! an xn , then f ÐkÑ (x) œ ! n(n 1)(n 2)â(n k 1)an xnk and f ÐkÑ (0) œ k! ak Ê ÐkÑ ak œ f k!(0) for k a nonnegative integer. Therefore, the coefficients of f(x) are identical with the corresponding coefficients in the Maclaurin series of f(x) and the statement follows. 52. Note: f even Ê f(x) œ f(x) Ê f w (x) œ f w (x) Ê f w (x) œ f w (x) Ê f w odd; f odd Ê f(x) œ f(x) Ê f w (x) œ f w (x) Ê f w (x) œ f w (x) Ê f w even; also, f odd Ê f(0) œ f(0) Ê 2f(0) œ 0 Ê f(0) œ 0 (a) If f(x) is even, then any odd-order derivative is odd and equal to 0 at x œ 0. Therefore, a" œ a$ œ a& œ á œ 0; that is, the Maclaurin series for f contains only even powers. (b) If f(x) is odd, then any even-order derivative is odd and equal to 0 at x œ 0. Therefore, a! œ a# œ a% œ á œ 0; that is, the Maclaurin series for f contains only odd powers. 53-58. Example CAS commands: Maple: f := x -> 1/sqrt(1+x); x0 := -3/4; x1 := 3/4; # Step 1: plot( f(x), x=x0..x1, title="Step 1: #53 (Section 10.9)" ); Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 626 Chapter 10 Infinite Sequences and Series # Step 2: P1 := unapply( TaylorApproximation(f(x), x = 0, order=1), x ); P2 := unapply( TaylorApproximation(f(x), x = 0, order=2), x ); P3 := unapply( TaylorApproximation(f(x), x = 0, order=3), x ); # Step 3: D2f := D(D(f)); D3f := D(D(D(f))); D4f := D(D(D(D(f)))); plot( [D2f(x),D3f(x),D4f(x)], x=x0..x1, thickness=[0,2,4], color=[red,blue,green], title="Step 3: #57 (Section 9.9)" ); c1 := x0; M1 := abs( D2f(c1) ); c2 := x0; M2 := abs( D3f(c2) ); c3 := x0; M3 := abs( D4f(c3) ); # Step 4: R1 := unapply( abs(M1/2!*(x-0)^2), x ); R2 := unapply( abs(M2/3!*(x-0)^3), x ); R3 := unapply( abs(M3/4!*(x-0)^4), x ); plot( [R1(x),R2(x),R3(x)], x=x0..x1, thickness=[0,2,4], color=[red,blue,green], title="Step 4: #53 (Section 10.9)" ); # Step 5: E1 := unapply( abs(f(x)-P1(x)), x ); E2 := unapply( abs(f(x)-P2(x)), x ); E3 := unapply( abs(f(x)-P3(x)), x ); plot( [E1(x),E2(x),E3(x),R1(x),R2(x),R3(x)], x=x0..x1, thickness=[0,2,4], color=[red,blue,green], linestyle=[1,1,1,3,3,3], title="Step 5: #53 (Section 10.9)" ); # Step 6: TaylorApproximation( f(x), view=[x0..x1,DEFAULT], x=0, output=animation, order=1..3 ); L1 := fsolve( abs(f(x)-P1(x))=0.01, x=x0/2 ); # (a) R1 := fsolve( abs(f(x)-P1(x))=0.01, x=x1/2 ); L2 := fsolve( abs(f(x)-P2(x))=0.01, x=x0/2 ); R2 := fsolve( abs(f(x)-P2(x))=0.01, x=x1/2 ); L3 := fsolve( abs(f(x)-P3(x))=0.01, x=x0/2 ); R3 := fsolve( abs(f(x)-P3(x))=0.01, x=x1/2 ); plot( [E1(x),E2(x),E3(x),0.01], x=min(L1,L2,L3)..max(R1,R2,R3), thickness=[0,2,4,0], linestyle=[0,0,0,2], color=[red,blue,green,black], view=[DEFAULT,0..0.01], title="#53(a) (Section 10.9)" ); abs(`f(x)`-`P`[1](x) ) <= evalf( E1(x0) ); # (b) abs(`f(x)`-`P`[2](x) ) <= evalf( E2(x0) ); abs(`f(x)`-`P`[3](x) ) <= evalf( E3(x0) ); Mathematica: (assigned function and values for a, b, c, and n may vary) Clear[x, f, c] f[x_]= (1 x)3/2 {a, b}= {1/2, 2}; pf=Plot[ f[x], {x, a, b}]; poly1[x_]=Series[f[x], {x,0,1}]//Normal poly2[x_]=Series[f[x], {x,0,2}]//Normal poly3[x_]=Series[f[x], {x,0,3}]//Normal Plot[{f[x], poly1[x], poly2[x], poly3[x]}, {x, a, b}, PlotStyle Ä {RGBColor[1,0,0], RGBColor[0,1,0], RGBColor[0,0,1], RGBColor[0,.5,.5]}]; Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.10 The Binomial Series 627 The above defines the approximations. The following analyzes the derivatives to determine their maximum values. f''[c] Plot[f''[x], {x, a, b}]; f'''[c] Plot[f'''[x], {x, a, b}]; f''''[c] Plot[f''''[x], {x, a, b}]; Noting the upper bound for each of the above derivatives occurs at x = a, the upper bounds m1, m2, and m3 can be defined and bounds for remainders viewed as functions of x. m1=f''[a] m2=-f'''[a] m3=f''''[a] r1[x_]=m1 x2 /2! Plot[r1[x], {x, a, b}]; r2[x_]=m2 x3 /3! Plot[r2[x], {x, a, b}]; r3[x_]=m3 x4 /4! Plot[r3[x], {x, a, b}]; A three dimensional look at the error functions, allowing both c and x to vary can also be viewed. Recall that c must be a value between 0 and x, so some points on the surfaces where c is not in that interval are meaningless. Plot3D[f''[c] x2 /2!, {x, a, b}, {c, a, b}, PlotRange Ä All] Plot3D[f'''[c] x3 /3!, {x, a, b}, {c, a, b}, PlotRange Ä All] Plot3D[f''''[c] x4 /4!, {x, a, b}, {c, a, b}, PlotRange Ä All] 10.10 THE BINOMIAL SERIES 1. (1 x)"Î# œ 1 "# x ˆ "# ‰ ˆ "# ‰ x# 2. (1 x)"Î$ œ 1 "3 x ˆ "3 ‰ ˆ 32 ‰ x# #! #! 3. (1 x)"Î# œ 1 "# (x) ˆ "# ‰ ˆ "# ‰ ˆ 3# ‰ x$ 3! ˆ 3" ‰ ˆ 32 ‰ ˆ 35 ‰ x$ #! #! ˆ x ‰# x ‰# ˆ (4)(3) 3 4 6. ˆ1 x3 ‰ œ 1 4 ˆ x3 ‰ #! "Î# 8. a1 x# b "Î$ œ 1 "# x$ œ 1 3" x# "Î# 9. ˆ1 1x ‰ œ 1 "# ˆ 1x ‰ ˆ "# ‰ ˆ "# ‰ (2x)# (2)(3) # # 5. ˆ1 x# ‰ œ 1 # ˆ x# ‰ #! 7. a1 x$ b 3! ˆ "# ‰ ˆ 3# ‰ (x)# 4. (1 2x)"Î# œ 1 "# (2x) ˆ "# ‰ ˆ 3# ‰ ˆ 5# ‰ (x)$ 3! Š "# ‹ Š "# ‹ Š 3# ‹ (2x)$ 3! $ 3! (4)(3)(2) ˆ x3 ‰ #! ˆ "3 ‰ ˆ 34 ‰ ax# b# #! ˆ "# ‰ ˆ "# ‰ ˆ 1x ‰# 5 $ á œ 1 3" x 9" x# 81 x á (2)(3)(4) ˆ x# ‰ ˆ "# ‰ ˆ 3# ‰ ax$ b# #! " $ á œ 1 "# x 8" x# 16 x á 3! $ á œ 1 x 12 x# 12 x$ á á œ 1 x 34 x# "# x$ (4)(3)(2)(1) ˆ x3 ‰ 4! ˆ "# ‰ ˆ 3# ‰ ˆ 5# ‰ ax$ b$ 3! ˆ 3" ‰ ˆ 34 ‰ ˆ 37 ‰ ax# b$ 3! ˆ "# ‰ ˆ "# ‰ ˆ 3# ‰ ˆ 1x ‰$ 3! 5 $ á œ 1 "# x 83 x# 16 x á 4 4 3 1 4 0 á œ 1 43 x 23 x2 27 x 81 x 5 * á œ 1 "# x$ 38 x' 16 x á ' á œ 1 3" x# 29 x% 14 81 x á " á œ 1 #"x 8x1 # 16x $ á Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 628 Chapter 10 Infinite Sequences and Series ˆ "3 ‰ ˆ 34 ‰ x# x 10. È œ xa1 xb"Î3 œ xŒ1 ˆc "3 ‰x 3 1x # $ ˆ 3" ‰ ˆ 34 ‰ ˆ 37 ‰ x$ #! 4 á œ x 13 x# 29 x3 14 81 x á 3! % 11. (1 x)% œ 1 4x (4)(3)x (4)(3)(2)x (4)(3)(2)x œ 1 4x 6x# 4x$ x% #! 3! 4! # # $ # $ ax b 12. a1 x# b œ 1 3x# (3)(2)#!ax b (3)(2)(1) œ 1 3x# 3x% x' 3! # $ 2x) 2x) 13. (1 2x)$ œ 1 3(2x) (3)(2)(# (3)(2)(1)( œ 1 6x 12x# 8x$ ! 3! % 14. ˆ1 x# ‰ œ 1 4 ˆ x# ‰ (4)(3) ˆ x# ‰ # #! (4)(3)(2) ˆ x# ‰ 3! $ (4)(3)(2)(1) ˆ x# ‰ 15. '0 sin x# dx œ '0 Šx# x3! x5! á ‹ dx œ ’ x3 7x†3! á “ 0Þ2 0Þ2 ' "! % 4! $ !Þ# ( ! ( " % œ 1 2x 32 x# "# x$ 16 x $ ¸ ’ x3 “ !Þ# ! ¸ 0.00267 with error kEk Ÿ (.2) 7†3! ¸ 0.0000003 16. '0 0Þ2 ecx " dx œ x '00 2 "x Š1 x x#! x3! x4! á 1‹ dx œ '00 2 Š1 x# x6 24x á ‹ dx Þ # # $ x œ ’x x4 18 á“ 17. '0 0Þ1 !Þ# ! $ Þ % % '00 1 Š1 x2 3x8 á ‹ dx œ ’x 10x á “ !Þ" ¸ [x]!Þ" ! ¸ 0.1 with error Þ " È1 x% dx œ !Þ#& $È $ ¸ 0.19044 with error kEk Ÿ (0.2) 96 ¸ 0.00002 % ) & ! & kEk Ÿ (0.1) 10 œ 0.000001 18. '! # 1 x# dx œ '0 0Þ25 # % $ & x á“ Š1 x3 x9 á ‹ dx œ ’x x9 45 & !Þ#& ! $ ¸ ’x x9 “ !Þ#& ! ¸ 0.25174 with error kEk Ÿ (0.25) 45 ¸ 0.0000217 19. '0 0Þ1 sin x x dx œ '00 1 Š1 x3! x5! x7! á ‹ dx œ ’x 3x†3! 5x†5! 7x†7! á “ !Þ" ¸ ’x 3x†3! 5x†5! “ !Þ" Þ # % ' $ & ( $ & ! ! 7 12 ¸ 0.0999444611, kEk Ÿ (0.1) 7†7! ¸ 2.8 ‚ 10 x x 20. '0 exp ax# b dx œ '0 Š1 x# x2! x3! x4! á ‹ dx œ ’x x3 10 42 á“ 0Þ1 0Þ1 % ' ) $ & ( !Þ" ! $ & 9 12 ¸ 0.0996676643, kEk Ÿ (0.1) 216 ¸ 4.6 ‚ 10 21. a1 x% b "Î# œ (1)"Î# ˆ "# ‰ ˆ "# ‰ ˆ 3# ‰ ˆ 5# ‰ 4! Š "# ‹ 1 (1)"Î# ax% b ˆ "# ‰ ˆ "# ‰ #! % % # (1)$Î# ax% b ) % 3! (1)&Î# ax% b ) "' "# "' & !Þ" ! 9 11 ¸ 0.100001, kEk Ÿ (0.1) 72 ¸ 1.39 ‚ 10 x‰ x 22. '0 ˆ 1 xcos dx œ '0 Š "# x4! x6! x8! 10! á ‹ dx ¸ ’ x# 3x†4! 5x†6! 7x†8! 9†x10! “ # 1 1 ¸ 0.4863853764, # % ' ) $ & 1 ( kEk Ÿ 11†112! ¸ 1.9 ‚ 1010 t t 23. '0 cos t# dt œ '0 Š1 t# 4! t6! á ‹ dt œ ’t 10 9t†4! 13t †6! á “ 1 $ (1)(Î# ax% b á œ 1 x# x8 x16 5x 128 á x Ê '0 Š1 x# x8 x16 5x 128 á ‹ dx ¸ ’x 10 “ 0Þ1 "# ˆ "# ‰ ˆ "# ‰ ˆ 3# ‰ % ) "# & * "$ " ! * ( x x ¸ ’x x3 10 42 “ " ! Ê kerrork 13"†6! ¸ .00011 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. !Þ" ! Section 10.10 The Binomial Series t t t 24. '0 cos Èt dt œ '0 Š1 #t 4! 6! 8! á ‹ dt œ ’t t4 3t†4! 4t†6! 5t†8! á “ 1 1 # $ % # $ % & Ê kerrork 5†"8! ¸ 0.000004960 t 25. F(x) œ '0 Št# 3! t5! t7! á ‹ dt œ ’ t3 7t†3! 11t †5! 15t †7! á “ x ' "! "% $ ( "" "& Ê kerrork 15"†7! ¸ 0.000013 x $ " ! ( "" ¸ x3 7x†3! 11x †5! ! t t 26. F(x) œ '0 Št# t% 2! 3! t4! t5! á ‹ dt œ ’ t3 t5 7t†2! 9t†3! 11t †4! 13t †5! á “ x $ ' & ( ) * "! "# $ & ( 629 * "" "$ "" x ! ¸ x3 x5 7x†2! 9x†3! 11x †4! Ê kerrork 13"†5! ¸ 0.00064 t 27. (a) F(x) œ '0 Št t3 t5 t7 á ‹ dt œ ’ t2 1t # 30 á “ ¸ x# 1x# Ê kerrork (0.5) 30 ¸ .00052 x $ & ( # # % x ' % ' # ' % ! ) $# (b) kerrork 33"†34 ¸ .00089 when F(x) ¸ x# 3x†4 5x†6 7x†8 á (1)"& 31x†32 28. (a) F(x) œ '0 Š1 2t t3 t4 á ‹ dt œ ’t 2t†2 3t†3 4t†4 5t†5 á “ ¸ x x## 3x# 4x# 5x# x # $ # $ % x & # $ % & ! ' Ê kerrork (0.5) 6# ¸ .00043 # $ % $" x (b) kerrork 32" # ¸ .00097 when F(x) ¸ x #x# 3x# 4x# á (1)$" 31 # # $ # 29. x"# aex (1 x)b œ x"# ŠŠ1 x x# x3! á ‹ 1 x‹ œ #" 3!x x4! á Ê lim xÄ0 ex (1 x) x# # œ lim Š "# 3!x x4! á ‹ œ "# xÄ0 $ # % # $ % $ & ( 2x 2x 30. "x aex ex b œ x" ’Š1 x x#! x3! x4! á ‹ Š1 x x#! x3! x4! á ‹“ œ "x Š2x 2x 3! 5! 7! á ‹ # % ' # ex e x 2x% 2x' œ x lim Š2 2x x 3! 5! 7! á ‹ œ 2 Ä_ 2x 2x œ 2 2x 3! 5! 7! á Ê lim xÄ0 # # # % ' # # % t t t t 31. t"% Š1 cos t t# ‹ œ t"% ’1 t# Š1 t# 4! 6! á ‹“ œ 4!" 6! 8! á Ê lim " cos t Š t# ‹ t% tÄ0 # % t t " œ lim Š 4!" 6! 8! á ‹ œ 24 tÄ0 $ $ $ & # $ % 32. )"& Š) )6 sin )‹ œ )"& Š) )6 ) )3! )5! á ‹ œ 5!" )7! )9! á Ê lim )Ä0 )% # sin ) ) Š )6 ‹ )& œ lim Š 5!" )7! 9! á ‹ œ 1#"0 )Ä0 $ & # % 33. y"$ ay tan" yb œ y"$ ’y Šy y3 y5 á ‹“ œ 3" y5 y7 á Ê lim yÄ0 # % y tan " y œ lim Š 3" y5 y7 á ‹ y$ yÄ0 œ "3 34. tanc" y sin y œ y$ cos y Ê lim yÄ0 y$ y& y$ y& Œy 3 5 á Œy 3! 5! á y$ cos y tanc" y sin y œ lim y$ cos y yÄ0 23y# " Œ 6 5! á cos y y$ œ Œ 6 23y& 5! á y$ cos y œ " Œ 6 23y# 5! á cos y œ "6 # # 35. x# Š1 e1Îx ‹ œ x# ˆ1 1 x"# #"x% 6x" ' á ‰ œ 1 #"x# 6x" % á Ê x lim x# Še1Îx 1‹ Ä_ ˆ1 #x" # 6x" % á ‰ œ 1 œ x lim Ä_ Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 630 Chapter 10 Infinite Sequences and Series 36. (x 1) sin ˆ x " 1 ‰ œ (x 1) Š x " 1 3!(x " 1)$ 5!(x " 1)& á ‹ œ 1 3!(x " 1)# 5!(x " 1)% á Ê x lim (x 1) sin ˆ x " 1 ‰ œ x lim Š1 3!(x " 1)# 5!(x " 1)% á ‹ œ 1 Ä_ Ä_ % 37. ' # % # % x x x x x x # Œx # 3 á Œ1 # 3 á Œ1 # 3 á ln a1 x# b ln a1 x# b œ œ Ê lim œ lim œ 2! œ 2 # # # % 1 cos x x Ä 0 1 cos x xÄ0 Š #"! x4! á‹ Š #"! x4! á‹ 1 Š1 x#! x4! á‹ (x 2)(x 2) # 38. lnx(x41) œ ’(x 2) (x c 2)# # x 2 œ lim (x c 2)$ á“ 3 # x Ä 2 ’1 x c# 2 (x 32) á“ œ x2 ’1 x # 2 Ê lim (x c 2)# á“ 3 x# 4 x Ä 2 ln (x 1) œ4 2 4 4 6 10 2 39. sin 3x2 œ 3x2 92 x6 81 40 x . . . and 1 cos 2x œ 2x 3 x 45 x . . . Ê lim sin 3x2 x Ä 0 1 cos 2x 10 8 3x2 92 x6 81 3 92 x4 81 40 x . . . 40 x . . . œ 32 2 2 4 4 2 2 x4 4 x6 . . . œ lim 2x 2 x x . . . xÄ0 xÄ0 3 45 3 45 œ lim 6 9 12 1 11 1 40. lna1 x3 b œ x3 x2 x3 x4 . . . and x sin x2 œ x3 16 x7 120 x 5040 x15 Þ Þ Þ Ê lim xÄ0 œ lim xÄ0 6 9 12 x3 x2 x3 x4 . . . 1 7 1 11 1 3 x 6 x 120 x 5040 x15 Þ Þ Þ 3 6 9 1 x2 x3 x4 . . . 1 4 1 8 1 1 6 x 120 x 5040 x12 Þ Þ Þ œ lim xÄ0 œ1 41. 1 1 21x 31x 41x Þ Þ Þ œ e1 œ e 3 4 5 3 2 1 1 1 4 1 42. ˆ 14 ‰ ˆ 14 ‰ ˆ 14 ‰ Þ Þ Þ œ ˆ 14 ‰ ”1 ˆ 14 ‰ ˆ 14 ‰ Þ Þ Þ • œ 64 1 1Î4 œ 64 3 œ 48 2 4 6 43. 1 432 2x 434 4x 436 6x Þ Þ Þ œ 1 21x ˆ 34 ‰ 41x ˆ 34 ‰ 61x ˆ 34 ‰ Þ Þ Þ œ cosˆ 34 ‰ 2 4 6 2 3 4 5 7 5 7 44. 12 2†122 3†123 4†124 Þ Þ Þ œ ˆ 12 ‰ 12 ˆ #1 ‰ 31 ˆ #1 ‰ 41 ˆ #1 ‰ Þ Þ Þ œ lnˆ1 21 ‰ œ lnˆ 23 ‰ 3 45. 13 313 3x 315 5x 317 7x Þ Þ Þ œ 13 31x ˆ 13 ‰ 51x ˆ 13 ‰ 71x ˆ 13 ‰ Þ Þ Þ œ sinˆ 13 ‰ œ 3 5 7 3 È3 2 46. 23 323 †3 325 †5 327 †7 Þ Þ Þ œ ˆ 23 ‰ 13 ˆ 23 ‰ 15 ˆ 23 ‰ 17 ˆ 32 ‰ Þ Þ Þ œ tan1 ˆ 32 ‰ 3 5 7 3 47. x3 x4 x5 x6 Þ Þ Þ œ x3 a1 x x2 x3 Þ Þ Þ b œ x3 ˆ 1 1 x ‰ œ 1 x x 2 2 4 4 6 6 48. 1 32xx 34xx 36xx Þ Þ Þ œ 1 21x a3xb2 41x a3xb4 61x a3xb6 Þ Þ Þ œ cosa3xb 2 3 3 49. x3 x5 x7 x9 Þ Þ Þ œ x3 Š1 x2 ax2 b ax2 b Þ Þ Þ ‹ œ x3 ˆ 1 +1x2 ‰ œ 1 +x x2 2 3 4 50. x2 2x3 22xx 23xx 24xx Þ Þ Þ œ x2 Š1 2x a2x2xb a2x3xb a2x4xb Þ Þ Þ ‹ œ x2 e2x 2 4 3 5 4 6 d d ˆ 1 ‰ 1 51. 1 2x 3x2 4x3 5x4 Þ Þ Þ œ dx a1 x x2 x3 x4 x5 Þ Þ Þ b œ dx 1 x œ a1 x b 2 52. 1 x2 x3 x4 x5 Þ Þ œ x1 Šx x2 x3 x4 x5 Þ Þ ‹ œ x1 lna1 xb œ lna1xxb 2 3 4 2 3 4 5 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. lnˆ1 x3 ‰ x sin x2 Section 10.10 The Binomial Series $ # % # $ % $ 631 & x‰ x x x x x x x x 53. ln ˆ 11 x œ ln (1 x) ln (1 x) œ Šx # 3 4 á ‹ Šx # 3 4 á ‹ œ 2 Šx 3 5 á ‹ # $ % " 54. ln (1 x) œ x x# x3 x4 á (1)n x á Ê kerrork œ ¹ (")n x ¹ œ n10 n when x œ 0.1; " " n10n 10) n 1 n Ê n10n 10) when n $ & ( n 1 n 8 Ê 7 terms * ) x 55. tan" x œ x x3 x5 x7 x9 á ("2n 1 " " #n1 10$ n 1 2n 1 á Ê kerrork œ ¹ (1)2nx1 n 1 2n 1 ¹ œ #n"1 when x œ 1; st Ê n 1001 # œ 500.5 Ê the first term not used is the 501 Ê we must use 500 terms & ( n1 2n1 * 56. tan" x œ x x3 x5 x7 x9 á (1)2nx1 $ ¸ 2n 1 ¸ œ x# á and n lim † 2n 1 ¹ œ x# n lim ¹x Ä _ 2n 1 x2n1 Ä _ #n 1 2n 1 _ Ê tan" x converges for kxk 1; when x œ 1 we have ! (2n1)1 which is a convergent series; when x œ 1 n nœ1 _ nc1 we have ! (2n1)1 which is a convergent series Ê the series representing tan" x diverges for kxk 1 nœ1 $ & ( * 57. tan" x œ x x3 x5 x7 x9 á (1)2n x1 n 1 2n 1 " ‰ á and when the series representing 48 tan" ˆ 18 has an error less than "3 † 10' , then the series representing the sum " ‰ " ‰ " ‰ 48 tan" ˆ 18 32 tan" ˆ 57 20 tan" ˆ #39 also has an error of magnitude less than 10' ; thus 2nc1 Š"‹ " kerrork œ 48 #18n 1 3†10 ' Ê n 4 using a calculator Ê 4 terms x x 58. ln (sec x) œ '0 tan t dt œ '0 Št t3 2t15 á ‹ dt ¸ x# 12 45 á x 59. (a) a1 x# b lim nÄ_ "Î# x # $ % & # ' % $ ' & ( 5x " ¸ 1 x# 3x8 5x x ¸ x x6 3x 16 Ê sin 40 112 ; Using the Ratio Test: 2nb3 (2n 1)(2n1) (2n 1)(2n 1)x 2†4†6â(2n)(2n ") # ¹ 12††34††56â ¹ ¹1 â(2n)(2n 2)(2n 3) † 1†3†5â(2n 1)x2nb1 ¹ 1 Ê x n lim Ä _ (2n 2)(2n 3) Ê kxk 1 Ê the radius of convergence is 1. See Exercise 69. (b) "Î# d " xb œ a1 x# b dx acos 60. (a) a1 t# b "Î# $ & ( $ & "‰ ˆ ¸ (1)"Î# ˆ "# ‰ (1)$Î# at# b # ˆ #3 ‰ (1) &Î# at# b #! # ˆ #" ‰ ˆ #3 ‰ ˆ #5 ‰ (1) (Î# at# b$ 3! 3†5t t 3t 5t x 3x 5x " œ 1 t# 23t # †2! 2$ †3! Ê sinh x ¸ '0 Š1 # 8 16 ‹ dt œ x 6 40 112 # % ' x # % ' $ & ( 3 " (b) sinh" ˆ 4" ‰ ¸ 4" 384 40,960 œ 0.24746908; the error is less than the absolute value of the first unused ( ( 5x 1 x 3x 5x Ê cos" x œ 1# sin" x ¸ 1# Šx x6 3x 40 112 ‹ ¸ # x 6 40 112 5x term, 112 , evaluated at t œ 4" since the series is alternating Ê kerrork 5 ˆ "4 ‰ 112 ( ¸ 2.725 ‚ 10' " d ˆ 1 ‰ " d # $ # $ 61. 1" x œ 1 (x) œ 1 x x x á Ê dx 1 x œ 1 x# œ dx a1 x x x á b œ 1 2x 3x# 4x$ á d ˆ " ‰ 2x d # % ' $ & 62. 1 " x# œ 1 x# x% x' á Ê dx 1 x# œ a1 x# b# œ dx a1 x x x á b œ 2x 4x 6x á Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 632 Chapter 10 Infinite Sequences and Series 8â(2n 2)†(2n) 63. Wallis' formula gives the approximation 1 ¸ 4 ’ 3†23††45††45††67††67†â (2n 1)†(2n 1) “ to produce the table n µ1 10 3.221088998 20 3.181104886 30 3.167880758 80 3.151425420 90 3.150331383 93 3.150049112 94 3.149959030 95 3.149870848 100 3.149456425 At n œ 1929 we obtain the first approximation accurate to 3 decimals: 3.141999845. At n œ 30,000 we still do not obtain accuracy to 4 decimals: 3.141617732, so the convergence to 1 is very slow. Here is a Maple CAS procedure to produce these approximations: pie := proc(n) local i,j; a(2) := evalf(8/9); for i from 3 to n do a(i) := evalf(2*(2*i2)*i/(2*i1)^2*a(i1)) od; [[j,4*a(j)] $ (j = n5 .. n)] end _ _ _ 64. (a) faxb œ 1 !ˆ mk ‰xk Ê f w axb œ !ˆ mk ‰k xk1 Ê a1 xb † f w axb œ a1 xb!ˆ mk ‰k xk1 kœ1 kœ1 _ _ kœ1 _ _ _ _ kœ2 kœ1 œ !ˆ mk ‰k xk 1 x † !ˆ mk ‰k xk1 œ !ˆ mk ‰k xk 1 !ˆ mk ‰k xk œ ˆ m1 ‰a1b x0 !ˆ mk ‰k xk1 !ˆ mk ‰k xk kœ1 kœ1 _ kœ1 _ œ m ! ˆ mk ‰k xk 1 ! ˆ mk ‰k xk kœ2 kœ1 _ kœ2 kœ1 _ m ‰ k Note that: ! ˆ mk ‰k xk 1 œ ! ˆ k 1 ak 1b x . _ w kœ1 _ _ _ Thus, a1 xb † f axb œ m ! ˆ m ‰k xk 1 ! ˆ m ‰k xk œ m ! ˆ m ‰ak 1b xk ! ˆ m ‰k xk kœ2 k kœ1 k _ _ kœ1 kœ1 kœ1 k1 kœ1 k k ! ’ˆˆ m ‰ak 1b ˆ m ‰k ‰xk “. ‰ ˆm‰ k œ m ! ’ˆ k m 1 ak 1b x k k x “ œ m k1 k m†am "bâam ak 1b 1b ‰ ˆm‰ Note that: ˆ k m ak 1b m†am "bâk!am k 1b k ak 1 b ! 1 ak 1b k k œ œ m†am "k!bâam kb m†am "bâk!am k 1b k œ m†am "bâk!am k 1b aam kb kb œ m m†am "bâk!am k 1b œ mˆ mk ‰. _ _ _ kœ1 kœ1 kœ1 m ‰ ! ’ˆmˆ m ‰ ‰xk “ œ m m!ˆ m ‰xk ˆm‰ ‰ k Thus, a1 xb † f w axb œ m ! ’ˆˆ k 1 ak 1b k k x “ œ m k k _ †faxb œ mŒ1 !ˆ mk ‰xk œ m † faxb Ê f w axb œ m a1xb if " x 1. kœ1 (b) Let gaxb œ a1 xbm faxb Ê gw axb œ ma1 xbm1 faxb a1 xbm f w axb †faxb m1 œ ma1 xbm1 faxb a1 xbm † m faxb a1 xbm1 † m † faxb œ 0. a1xb œ ma1 xb _ (c) gw axb œ 0 Ê gaxb œ c Ê a1 xbm faxb œ c Ê faxb œ a1cxbcm œ ca1 xbm . Since faxb œ 1 !ˆ mk ‰xk kœ1 _ Ê fa0b œ 1 !ˆ mk ‰a0bk œ 1 0 œ 1 Ê ca1 0bm œ 1 Ê c œ 1 Ê faxb œ a1 xbm . kœ1 65. a1 x# b "Î# œ a1 ax# bb "Î# ˆ "# ‰ ˆ 3# ‰ ˆ 5# ‰ (1) (Î# ax# b$ 3! œ (1)"Î# ˆ "# ‰ (1)$Î# ax# b ˆ "# ‰ ˆ 3# ‰ (1)c&Î# ax# b# _ #! 1†3†5x ! 1†3†5ân(2n1)x á œ 1 x# 12†#3x †#! 2$ †3! á œ 1 # †n! # % ' 2n nœ1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Section 10.10 The Binomial Series Ê sin" x œ '0 a1 t# b x 633 1†3†5â(2n 1)x 1)x dt œ '0 Œ1 ! 1†3†5â#(2n n †n! dt œ x ! #†4â(2n)(2n 1) , _ x "Î# _ 2n nœ1 2nb1 nœ1 where kxk 1 _ dt 66. ctan" td x œ 1# tan" x œ 'x _ _ 1 t# _ _ " Š 1# ‹ œ 'x – t " — dt œ 'x 1Š ‹ t# œ 'x ˆ t"# t"% t"' t") á ‰ dt œ lim bÄ_ t# ˆ1 t"# t"% t"' á ‰ dt "t 3t"$ 5t"& 7t"( á ‘ b œ x" 3x" $ 5x" & 7x" ( á x Ê tan" x œ 1# "x 3x" $ 5x" & á , x 1; ctan" td c_ œ tan" x 1# œ ' _ 1 dt t# x x œ lim b Ä _ "t 3t"$ 5t"& 7t"( á ‘ x œ x" 3x" $ 5x" & 7x" ( á Ê tan" x œ 1# x" 3x" $ 5x" & á , b x 1 67. (a) ei1 œ cos (1) i sin (1) œ 1 i(0) œ 1 (b) ei1Î4 œ cos ˆ 14 ‰ i sin ˆ 14 ‰ œ È" Èi œ Š È" ‹ (1 i) 2 2 2 (c) ei1Î2 œ cos ˆ 1# ‰ i sin ˆ 1# ‰ œ 0 i(1) œ i 68. ei) œ cos ) i sin ) Ê ei) œ ei()) œ cos ()) i sin ()) œ cos ) i sin ); i) ei) ei) œ cos ) i sin ) cos ) i sin ) œ 2 cos ) Ê cos ) œ e #e i) i) e e ci) ; i) ci) sin ) œ e #ie œ cos ) i sin ) (cos ) i sin )) œ 2i sin ) Ê # $ % )) )) )) 69. ex œ 1 x x#! x3! x4! á Ê ei) œ 1 i) (i2! (i3! (i4! á and # $ % # $ % # $ % )) )) ei) œ 1 i) (2!i)) (3!i)) (4!i)) á œ 1 i) (i#)!) (i3! (i4! á # $ % # $ % )) )) (i4! á‹ Š1 i) (i)) (i)) (i)) á‹ Š1 i) (i#)!) (i3! i) ci) 3! 4! #! Ê e #e œ # % ' œ 1 )#! )4! )6! á œ cos ); # $ # % # $ % )) )) (i4! á‹ Š1 i) (i)) (i)) (i)) á‹ Š1 i) (i#)!) (i3! 3! 4! #! ei) eci) œ #i $ & ( œ ) )3! )5! )7! á œ sin ) #i 70. ei) œ cos ) i sin ) Ê ei) œ eiÐ)Ñ œ cos ()) i sin ()) œ cos ) i sin ) i) (a) ei) ei) œ (cos ) i sin )) (cos ) i sin )) œ 2 cos ) Ê cos ) œ e #e (b) ei) ei) œ (cos ) i sin )) (cos ) i sin )) œ 2i sin ) Ê i sin ) œ # $ % $ & ci) œ cosh i) e eci) i) 2 œ sinh i) ( 71. ex sin x œ Š1 x x#! x3! x4! á ‹ Šx x3! x5! x7! á ‹ " & œ (1)x (1)x# ˆ 6" #" ‰ x$ ˆ 6" 6" ‰ x% ˆ 1#"0 1"# #"4 ‰ x& á œ x x# 3" x$ 30 x á ; ex † eix œ eÐ1iÑx œ ex (cos x i sin x) œ ex cos x i aex sin xb Ê ex sin x is the series of the imaginary part _ # $ % of eÐ1iÑx which we calculate next; eÐ1iÑx œ ! (xn!ix) œ 1 (x ix) (x #!ix) (x 3!ix) (x 4!ix) á n nœ0 œ 1 x ix #"! a2ix# b 3!" a2ix$ 2x$ b 4!" a4x% b 5!" a4x& 4ix& b 6!" a8ix' b á Ê the imaginary part " & " ' of eÐ1iÑx is x #2! x# 3!2 x$ 5!4 x& 6!8 x' á œ x x# "3 x$ 30 x 90 x á in agreement with our product calculation. The series for ex sin x converges for all values of x. d ˆ ÐaibÑ ‰ d 72. dx e œ dx ceax (cos bx i sin bx)d œ aeax (cos bx i sin bx) eax (b sin bx bi cos bx) œ aeax (cos bx i sin bx) bieax (cos bx i sin bx) œ aeÐaibÑx ibeÐaibÑx œ (a ib)eÐaibÑx Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 634 Chapter 10 Infinite Sequences and Series 73. (a) ei)" ei)# œ (cos )" i sin )" )(cos )# i sin )# ) œ (cos )" cos )# sin )" sin )# ) i(sin )" cos )# sin )# cos )" ) œ cos()" )# ) i sin()" )# ) œ eiÐ)" )# Ñ " " ) i sin ) ‰ (b) ei) œ cos()) i sin()) œ cos ) i sin ) œ (cos ) i sin )) ˆ cos cos ) i sin ) œ cos ) i sin ) œ ei) 74. aa# bib# eÐabiÑx C" iC# œ ˆ aa# bib# ‰ eax (cos bx i sin bx) C" iC# ax œ a# e b# (a cos bx ia sin bx ib cos bx b sin bx) C" iC# ax œ a# e b# [(a cos bx b sin bx) (a sin bx b cos bx)i] C" iC# œ e (a cosa#bxb#b sin bx) C" ie (a sina#bxb#b cos bx) iC# ; ax ax eÐabiÑx œ eax eibx œ eax (cos bx i sin bx) œ eax cos bx ieax sin bx, so that given ' eÐabiÑx dx œ aa bib eÐabiÑx C" iC# we conclude that ' eax cos bx dx œ e (a cosa bxb b sin bx) C" and ' eax sin bx dx œ e (a sinabxbb cos bx) C# ax # # # # ax # # CHAPTER 10 PRACTICE EXERCISES 1. converges to 1, since n lim a œ n lim Š1 (n1) ‹ œ 1 Ä_ n Ä_ n 2 2. converges to 0, since 0 Ÿ an Ÿ È2n , n lim 0 œ 0, n lim œ 0 using the Sandwich Theorem for Sequences Ä_ Ä _ Èn ˆ 1 2n2 ‰ œ lim ˆ #"n 1‰ œ 1 3. converges to 1, since n lim a œ n lim Ä_ n Ä_ nÄ_ n 4. converges to 1, since n lim a œ n lim c1 (0.9)n d œ 1 0 œ 1 Ä_ n Ä_ ™ œ e0ß 1ß 0ß 1ß 0ß 1ß á f 5. diverges, since ˜sin n1 # 6. converges to 0, since {sin n1} œ {0ß 0ß 0ß á } # ln n 7. converges to 0, since n lim a œ n lim œ 2 n lim Ä_ n Ä_ n Ä_ Š "n ‹ 1 Š 2n 2b 1 ‹ ln (2n") 8. converges to 0, since n lim a œ n lim œ n lim n Ä_ n Ä_ Ä_ 1 1Š "n ‹ ˆ n nln n ‰ œ lim 9. converges to 1, since n lim a œ n lim Ä_ n Ä_ nÄ_ $ œ0 1 ln a2n 1b 10. converges to 0, since n lim a œ n lim œ n lim n Ä_ n Ä_ Ä_ Š œ0 œ1 6n# ‹ 2n$ 1 1 12n 2 œ n lim œ n lim œ0 Ä _ 6n# Ä_ n n ˆ n n 5 ‰ œ lim Š1 (n5) ‹ œ ec5 by Theorem 5 11. converges to ec5 , since n lim a œ n lim Ä_ n Ä_ nÄ_ n ˆ1 n" ‰ 12. converges to "e , since n lim a œ n lim Ä_ n Ä_ n " " œ n lim n œ e by Theorem 5 Ä _ ˆ1 " ‰ n 1 În 3 œ n lim œ 13 œ 3 by Theorem 5 Ä _ n1În 1În 3 œ n lim œ 11 œ 1 by Theorem 5 Ä _ n1În ˆ3 ‰ 13. converges to 3, since n lim a œ n lim Ä_ n Ä_ n ˆ3‰ 14. converges to 1, since n lim a œ n lim Ä_ n Ä_ n cn 1În Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Chapter 10 Practice Exercises 1În 2 1 15. converges to ln 2, since n lim a œ n lim n a21În 1b œ n lim œ n lim " Ä_ n Ä_ Ä_ Ä_ – Š 21În ln 2‹ n# Š n#" ‹ Šn‹ — 635 œ n lim 21În ln 2 Ä_ œ 2! † ln 2 œ ln 2 2 n È 16. converges to 1, since n lim a œ n lim 2n 1 œ n lim exp Š ln (2nn 1) ‹ œ n lim exp Œ 2n1b 1 œ e! œ 1 Ä_ n Ä_ Ä_ Ä_ (n 1)! 17. diverges, since n lim a œ n lim œ n lim (n 1) œ _ n! Ä_ n Ä_ Ä_ (4) 18. converges to 0, since n lim a œ n lim œ 0 by Theorem 5 Ä_ n Ä _ n! n Š"‹ Š"‹ # # " 19. (2n 3)(2n 1) œ #n 3 2n 1 Ê sn œ – Š "# ‹ 3 Š "# ‹ 5 —– Š "# ‹ 5 Š "# ‹ 7 Š "# ‹ Š "# ‹ — á – #n 3 2n 1 — œ Š "# ‹ 3 Š "‹ # 2n 1 Š "‹ # " " Ê n lim s œ n lim 2n 1— œ 6 Ä_ n Ä _ –6 20. n(n2 1) œ n2 n 2 1 Ê sn œ ˆ #2 23 ‰ ˆ 32 42 ‰ á ˆ n2 n 2 1 ‰ œ #2 n 2 1 Ê n lim s Ä_ n ˆ1 n 2 1 ‰ œ 1 œ n lim Ä_ 9 3 3 3 ‰ 3 ‰ ˆ3 3‰ ˆ3 3‰ ˆ3 ˆ 3 21. (3n 1)(3n 2) œ 3n 1 3n 2 Ê sn œ # 5 5 8 8 11 á 3n 1 3n 2 ˆ 3 3n 3 2 ‰ œ #3 œ 3# 3n3# Ê n lim s œ n lim Ä_ n Ä_ # 8 2 2 2 ‰ 2 ‰ 2 ‰ ˆ 92 13 22. (4n 3)(4n ˆ 132 17 ˆ 172 21 á ˆ 4n2 3 4n 2 1 ‰ 1) œ 4n 3 4n 1 Ê sn œ ˆ 29 4n21 ‰ œ 29 œ 29 4n21 Ê n lim s œ n lim Ä_ n Ä_ _ _ nœ0 nœ0 23. ! en œ ! e"n , a convergent geometric series with r œ "e and a œ 1 Ê the sum is _ _ nœ1 nœ0 " œ e e 1 1 Š "e ‹ ‰ a convergent geometric series with r œ "4 and a œ 43 Ê the sum is 24. ! (1)n 43n œ ! ˆ 34 ‰ ˆ " 4 n ˆ 34 ‰ œ 35 1ˆ c4" ‰ 25. diverges, a p-series with p œ "# _ _ nœ1 nœ1 26. ! n5 œ 5 ! "n , diverges since it is a nonzero multiple of the divergent harmonic series " 27. Since f(x) œ x"Î# Ê f w (x) œ #x"$Î# 0 Ê f(x) is decreasing Ê an1 an , and _ _ 1 lim a œ lim œ 0, the n Ä _ n n Ä _ Èn ) ! " diverges, the given series converges conditionally. series ! (" Èn converges by the Alternating Series Test. Since Èn nœ1 n nœ1 28. converges absolutely by the Direct Comparison Test since #n" $ n"$ for n 1, which is the nth term of a convergent p-series Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 636 Chapter 10 Infinite Sequences and Series 29. The given series does not converge absolutely by the Direct Comparison Test since ln (n" 1) n " 1 , which is " the nth term of a divergent series. Since f(x) œ ln (x" 1) Ê f w (x) œ (ln (x 1)) # (x 1) 0 Ê f(x) is decreasing " Ê an1 an , and n lim a œ n lim œ 0, the given series converges conditionally by the Alternating Ä_ n Ä _ ln (n 1) Series Test. _ 30. '2 " x(ln x)# dx œ b lim Ä_ '2b x(ln" x) dx œ lim c(ln x)" d b2 œ lim ˆ ln"b ln"2 ‰ œ ln"# Ê the series # bÄ_ bÄ_ converges absolutely by the Integral Test 31. converges absolutely by the Direct Comparison Test since lnn$n nn$ œ n"# , the nth term of a convergent p-series n n 32. diverges by the Direct Comparison Test for en n Ê ln ên ‰ ln n Ê nn ln n Ê ln nn ln (ln n) Ê n ln n ln (ln n) Ê lnln(lnnn) n" , the nth term of the divergent harmonic series 33. n lim Ä_ Š È "# n n Š n"# ‹ 1 ‹ # n œ Én lim œ È1 œ 1 Ê converges absolutely by the Limit Comparison Test Ä _ n# 1 $ 34. Since f(x) œ x$3x 1 Ê f w (x) œ 3xaxa$21xb# b 0 when x # # 2 Ê an1 an for n 3n 2 and n lim œ 0, the Ä _ n$ 1 series converges by the Alternating Series Test. The series does not converge absolutely: By the Limit # Comparison Test, n lim Ä_ Š n$3n 1 ‹ ˆ n" ‰ $ 3n œ n lim œ 3. Therefore the convergence is conditional. Ä _ n$ 1 n2 35. converges absolutely by the Ratio Test since n lim œ01 ’ n2 † n! “ œ n lim Ä _ (n 1)! n 1 Ä _ (n 1)# # (") an 1b 36. diverges since n lim a œ n lim does not exist Ä_ n Ä _ 2n# n 1 n nb1 3 37. converges absolutely by the Ratio Test since n lim † n! “ œ n lim œ01 ’ 3 Ä _ (n 1)! 3n Ä _ n1 n 2 3 6 n È É 38. converges absolutely by the Root Test since n lim an œ n lim œ01 nn œ n lim Ä_ Ä_ Ä_ n n n 39. converges absolutely by the Limit Comparison Test since n lim Ä_ 40. converges absolutely by the Limit Comparison Test since n lim Ä_ nb1 " Š $Î# ‹ n Š Èn(n "1)(n Š n"# ‹ Š È "# n n ‹ 2) n(n 1)(n 2) œ Én lim œ1 n$ Ä_ # 1 ‹ # n an 1 b œ Én lim œ1 n% Ä_ kx 4 k ˆ n ‰ 1 Ê kx 3 4k 1 41. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x 4) † n3 ¹ 1 Ê 3 n lim Ä_ Ä _ (n 1)3nb1 (x 4)n Ä _ n1 n _ _ 1) 3 Ê kx 4k 3 Ê 3 x 4 3 Ê 7 x 1; at x œ 7 we have ! (n3 œ ! ("n ) , the alternating n _ nœ1 _ n n n nœ1 3 ! " , the divergent harmonic series harmonic series, which converges conditionally; at x œ 1 we have! n3 n œ n nœ1 n nœ1 (a) the radius is 3; the interval of convergence is 7 Ÿ x 1 (b) the interval of absolute convergence is 7 x 1 (c) the series converges conditionally at x œ 7 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Chapter 10 Practice Exercises " 42. n lim œ 0 1, which holds for all x ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x1) † (2n1)! ¹ 1 Ê (x 1)# n lim Ä_ Ä _ (2n1)! (x1)2nc2 Ä _ (#n)(2n1) 2n (a) the radius is _; the series converges for all x (b) the series converges absolutely for all x (c) there are no values for which the series converges conditionally nb1 n 43. n lim 1 Ê k3x 1k 1 ¹ uunbn 1 ¹ 1 Ê n lim ¹ (3x(n1)1)# † (3x n 1)n ¹ 1 Ê k3x 1k n lim Ä_ Ä_ Ä _ (n 1)# # # _ nc1 _ 2nc1 Ê 1 3x 1 1 Ê 0 3x 2 Ê 0 x 23 ; at x œ 0 we have ! (1) n# (1) œ ! ("n)# n nœ1 nœ1 _ œ ! n"# , a nonzero constant multiple of a convergent p-series, which is absolutely convergent; at x œ 23 we nœ1 _ _ have ! (1)n# (1) œ ! ("n)# n 1 n nœ1 n 1 , which converges absolutely nœ1 (a) the radius is "3 ; the interval of convergence is 0 Ÿ x Ÿ 23 (b) the interval of absolute convergence is 0 Ÿ x Ÿ 23 (c) there are no values for which the series converges conditionally 44. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ n 2 † (2x 2nb1)1 Ä_ Ä _ 2n 3 nb1 k2x 1k 2 1 † 2n lim ¸ n 2 † 2n " ¸ 1 n 1 † (2x 1)n ¹ 1 Ê 2 n Ä _ 2n 3 n 1 n Ê k2x # 1k (1) 1 Ê k2x 1k 2 Ê 2 2x 1 2 Ê 3 2x 1 Ê 3# x "# ; at x œ 3# we have _ _ ! n 1 † (2) œ ! (") (n1) which diverges by the nth-Term Test for Divergence since n nœ1 n 2n 1 # nœ1 n 2n 1 _ _ n1 2 ! n " , which diverges by the nth-Term Test lim ˆ n 1 ‰ œ #" Á 0; at x œ #" we have ! 2n 1 † #n œ 2n 1 n Ä _ 2n 1 n nœ1 nœ1 (a) the radius is 1; the interval of convergence is 3# x "# (b) the interval of absolute convergence is 3# x "# (c) there are no values for which the series converges conditionally nb1 ¸ˆ n ‰ ˆ n " 1 ‰¸ 1 Ê kxek lim ˆ n " 1 ‰ 1 45. n lim † n ¹ 1 Ê kxk n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ x Ä_ Ä _ (n 1)nb1 xn Ä _ n1 nÄ_ n n Ê kxek † 0 1, which holds for all x (a) the radius is _; the series converges for all x (b) the series converges absolutely for all x (c) there are no values for which the series converges conditionally nb1 Èn n 46. n lim † xn ¹ 1 Ê kxk n lim 1 Ê kxk 1; when x œ 1 we have ¹ uunbn 1 ¹ 1 Ê n lim ¹ x Ä_ Ä _ Èn 1 Ä _ Én1 _ _ ! (È1) , which converges by the Alternating Series Test; when x œ 1 we have ! È" , a divergent p-series nœ1 n n nœ1 n (a) the radius is 1; the interval of convergence is 1 Ÿ x 1 (b) the interval of absolute convergence is 1 x 1 (c) the series converges conditionally at x œ 1 2nb1 47. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (n 32)x nb1 Ä_ Ä_ _ _ nœ1 nœ1 # 3 x ˆ n 2 ‰ 1 Ê È3 x È3; † (n 1)x 2n 1 ¹ 1 Ê 3 n lim Ä _ n1 n the series ! nÈ31 and ! nÈ31 , obtained with x œ „ È3, both diverge (a) the radius is È3; the interval of convergence is È3 x È3 (b) the interval of absolute convergence is È3 x È3 (c) there are no values for which the series converges conditionally Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 637 638 Chapter 10 Infinite Sequences and Series 2nb3 48. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ (x 2n1)x 3 Ä_ Ä_ 1 # ˆ 2n 1 ‰ 1 Ê (x 1)# (1) 1 † (x 2n 1)2nb1 ¹ 1 Ê (x 1) n lim Ä _ 2n 3 _ Ê (x 1)# 1 Ê kx 1k 1 Ê 1 x 1 1 Ê 0 x 2; at x œ 0 we have ! (1)#n(1)1 n 2nb1 nœ1 _ œ ! (1) nœ1 _ 3nb1 2n 1 _ œ ! (1) nœ1 nc1 2n 1 which converges conditionally by the Alternating Series Test and the fact _ 2nb1 that ! 2n " 1 diverges; at x œ 2 we have ! (1)2n (1) 1 nœ1 n nœ1 _ (1) œ ! 2n 1 , which also converges conditionally n nœ1 (a) the radius is 1; the interval of convergence is 0 Ÿ x Ÿ 2 (b) the interval of absolute convergence is 0 x 2 (c) the series converges conditionally at x œ 0 and x œ 2 (n 1)x 49. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ cschcsch (n)xn Ä_ Ä_ c" nb1 Š enb1 c ecnc1 ‹ 1 ¹ 1 Ê kxk n lim Ä_ » ˆ 2 ‰ » 2 en c ecn _ kx k Ê kxk n lim ¹ e1 ee 2n 2 ¹ 1 Ê e 1 Ê e x e; the series ! a „ ebn csch n, obtained with x œ „ e, Ä_ 2n 1 nœ1 both diverge since n lim a „ e) csch n Á 0 Ä_ n (a) the radius is e; the interval of convergence is e x e (b) the interval of absolute convergence is e x e (c) there are no values for which the series converges conditionally nb1 c2nc2 c2n (n 1) 50. n lim † 1 e ¹ 1 Ê kxk 1 ¹ uunbn 1 ¹ 1 Ê n lim ¹ x xncoth ¹1e coth (n) ¹ 1 Ê kxk n lim Ä_ Ä_ Ä _ 1 ec2nc2 1 ec2n _ Ê 1 x 1; the series ! a „ 1bn coth n, obtained with x œ „ 1, both diverge since n lim a „ 1bn coth n Á 0 Ä_ nœ1 (a) the radius is 1; the interval of convergence is 1 x 1 (b) the interval of absolute convergence is 1 x 1 (c) there are no values for which the series converges conditionally 51. The given series has the form 1 x x# x$ á (x)n á œ 1 " x , where x œ 4" ; the sum is 1 "ˆ " ‰ œ 54 4 # $ n $ & 2n 1 # % 2n 52. The given series has the form x x# x3 á (1)n1 xn á œ ln (1 x), where x œ 23 ; the sum is ln ˆ 53 ‰ ¸ 0.510825624 53. The given series has the form x x3! x5! á (1)n (2nx 1)! á œ sin x, where x œ 1; the sum is sin 1 œ 0 x 54. The given series has the form 1 x2! x4! á (1)n (2n)! á œ cos x, where x œ 13 ; the sum is cos 13 œ "# # # 55. The given series has the form 1 x x2! x3! á xn! á œ ex , where x œ ln 2; the sum is eln Ð2Ñ œ 2 $ n & x " 56. The given series has the form x x3 x5 á (1)n (2n x, where x œ È"3 ; the sum is 1) á œ tan 2n 1 tan" Š È"3 ‹ œ 16 57. Consider 1 " 2x as the sum of a convergent geometric series with a œ 1 and r œ 2x Ê 1 " 2x _ _ nœ0 nœ0 œ 1 (2x) (2x)# (2x)$ á œ ! (2x)n œ ! 2n xn where k2xk 1 Ê kxk "# Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Chapter 10 Practice Exercises 639 58. Consider 1 " x$ as the sum of a convergent geometric series with a œ 1 and r œ x$ Ê 1 " x$ œ 1 a"x$ b # _ $ œ 1 ax$ b ax$ b ax$ b á œ ! (1)n x3n where kx$ k 1 Ê kx$ k 1 Ê kxk 1 nœ0 _ _ n 2nb1 2nb1 1) (1x) 59. sin x œ ! ((2n1)x 1)! Ê sin 1x œ ! ((2n 1)! nœ0 n nœ0 _ nœ0 2nb1 _ n 2n nœ0 _ Ê cos ˆx5Î3 ‰ œ ! nœ0 (1)n ˆx5Î3 ‰ (2n)! 2n _ (1)n Š x3 ‹ È Ê cosŠ Èx 5 ‹ œ ! _ _ ˆ 1 x ‰n n 2n nœ0 3 5 (2n)! nœ0 63. ex œ ! xn! Ê eÐ1xÎ2Ñ œ ! n nœ0 nœ0 _ _ n! œ! # n _ # n œ0 n 2nb1 2nb1 nœ0 _ n 10nÎ3 x œ ! (1)(#n)! nœ0 2n _ 1) x 62. cos x œ ! ((2n)! _ _ œ ! (32n1)b12(#n x 1)! (2n 1)! nœ0 n 2nb1 2nb1 nœ0 _ (1)n Š 2x ‹ 3 n 2nb1 ! 60. sin x œ ! ((2n1)x 1)! Ê sin 2x 3 œ 1) x 61. cos x œ ! ((2n)! _ œ ! (1)(#1n 1)!x _ x œ ! (5n1)(#n)! n 6n nœ0 1 n xn #n n! 64. ex œ ! xn! Ê ex œ ! an!x b œ ! (1)n! x # n nœ0 nœ0 nœ0 65. f(x) œ È3 x# œ a3 x# b Ê f www (x) œ 3x$ a3 x# b 3 9 f www (1) œ 32 38 œ 32 n 2n "Î# Ê f w (x) œ x a3 x# b &Î# "Î# Ê f ww (x) œ x# a3 x# b $Î# a3 x# b "Î# $Î# 3x a3 x# b ; f(1) œ 2, f w (1) œ "# , f ww (1) œ "8 #" œ 38 , # $ Ê È3 x# œ 2 (x 1) 3(x$ 1) 9(x& 1) á 2†1! 2 †2! 2 †3! 66. f(x) œ 1 " x œ (1 x)" Ê f w (x) œ (1 x)# Ê f ww (x) œ 2(1 x)$ Ê f www (x) œ 6(1 x)% ; f(2) œ 1, f w (2) œ 1, f ww (2) œ 2, f www (2) œ 6 Ê 1" x œ 1 (x 2) (x 2)# (x 2)$ á 67. f(x) œ x " 1 œ (x 1)" Ê f w (x) œ (x 1)# Ê f ww (x) œ 2(x 1)$ Ê f www (x) œ 6(x 1)% ; f(3) œ "4 , f w (3) œ 4"# , f ww (3) œ 42$ , f www (2) œ 4%6 Ê x" 1 œ "4 4"# (x 3) 4"$ (x 3)# 4"% (x 3)$ á 68. f(x) œ "x œ x" Ê f w (x) œ x# Ê f ww (x) œ 2x$ Ê f www (x) œ 6x% ; f(a) œ "a , f w (a) œ a"# , f ww (a) œ a2$ , f www (a) œ a%6 Ê "x œ "a a"# (x a) a"$ (x a)# a"% (x a)$ á 69. '0 exp ax$ b dx œ '0 Š1 x$ x2! x3! x4! á ‹ dx œ ’x x4 7x†2! 10x †3! 13x †4! á “ 1Î2 1Î2 ' * "# % ( "! "$ " " " ¸ "# #%"†4 #( †"7†2! 2"! †10 †3! 2"$ †13†4! 2"' †16†5! ¸ 0.484917143 "Î# ! 70. '0 x sin ax$ b dx œ '0 x Šx$ x3! x5! x7! x9! á ‹ dx œ '0 Šx% x3! x5! x7! x9! á ‹ dx 1 1 & "" "( * #$ "& #" #( 1 "! "' ## #) " #* œ ’ x5 11x †3! 17x †5! 23x †7! 29x †9! á “ ¸ 0.185330149 ! 71. '1 1Î2 tanc" x dx œ x "Î# x '11Î2 Š1 x3 x5 x7 x9 x11 á ‹ dx œ ’x x9 25x 49x 81x 121 á“ # % ' ) "! $ & ( * "" ! ¸ "# 9†"2$ 5#"†#& 7#"†#( 9#"†2* 11#"†2"" 13#"†2"$ 15#"†2"& 17#"†#"( 19#"†#"* 21#"†##" ¸ 0.4872223583 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 640 Chapter 10 Infinite Sequences and Series 72. '0 1Î64 tan " x Èx dx œ '01Î64 È"x Šx x3$ x5& x7( á ‹ dx œ '01Î64 ˆx"Î# 3" x&Î# 5" x*Î# 7" x"$Î# á ‰ dx "Î'% 2 ""Î# 2 œ 23 x$Î# #21 x(Î# 55 x 105 x"&Î# á ‘ ! $ & œ ˆ 3†28$ 212†8( 55†28"" 1052†8"& á ‰ ¸ 0.0013020379 # 7 Šx x x á ‹ % 7 Š1 x x á ‹ 3! 5! 3! 5! 7 sin x œ lim œ lim œ 7# 2x # # $ $ # $ # x Ä 0 e 1 x Ä 0 Š2x 2 2!x 2 3!x á‹ x Ä 0 Š2 2#!x 2 3!x á‹ 73. lim 74. # $ # $ Š1 ) )#! )3! á‹ Š1 ) )#! )3! á‹ 2) ) c) lim e )e sin)2) œ lim )Ä0 )Ä0 $ & ) Š) )3! )5! á‹ $ & $ & 2 Š )3! )5! á‹ œ lim )Ä0 Š )3! )5! á‹ # œ lim )Ä0 2 Š 3!" )5! á‹ œ2 # Š 3!" )5! á‹ # 75. # lim ˆ " t"# ‰ œ lim t 2t# (12 2coscost) t œ lim t Ä 0 # 2 cos t tÄ0 tÄ0 % t t# 2 2 Œ1 t# 4! á # % t 2t# Š1 1 t# 4! á‹ % œ lim tÄ0 ' t 2 Œ 4! t6! á ' Št% 2t4! á‹ # œ lim tÄ0 76. lim " 2 Š 4! t6! á‹ # Š1 2t4! á‹ Š sinh h ‹ cos h hÄ0 œ lim h# œ 1"# h# œ lim h% h# h# hÄ0 h# h# h% h% h' h' Œ #! 3! 5! 4! 6! 7! á h# hÄ0 h% Œ1 3! 5! á Œ1 #! 4! á # # % % Š #"! 3!" h5! h4! h6! h7! á ‹ œ 3" œ lim hÄ0 % % 1 Š1 z# z3 á‹ Šz# z3 á‹ " cos# z œ lim œ lim # $ $ & # $ % z Ä 0 ln (1 z) sin z z Ä 0 Šz z# z3 á‹ Šz z3! z5! á‹ z Ä 0 Š z# 2z3 z4 á‹ 77. lim # œ lim Š1 z3 á‹ # z z Ä 0 Š "# 2z 3 4 á‹ œ 2 y# y# y# œ lim œ lim # % ' # % ' # cos y cosh y y y y y y y 2y 2y' yÄ0 y Ä 0 Œ1 á Œ1 á y Ä 0 Œ 4! 6! 4! 6! # #! # 6! á 78. lim œ lim " y Ä 0 Œ1 2y% á 6! œ 1 $ 79. lim ˆ sinx$3x xr# s‰ œ lim – xÄ0 xÄ0 & (3x) Š3x (3x) 6 120 á‹ x$ # r xr# s— œ lim Š x3# #9 81x 40 á x# s‹ œ 0 xÄ0 Ê xr# x3# œ 0 and s 9# œ 0 Ê r œ 3 and s œ 9# 80. The approximation sin x ¸ 6 6xx# is better than sin x ¸ x. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Chapter 10 Practice Exercises (3n 1)(3n 2)x 81. n lim ¹ 2†5†8#â †4†6â(2n)(2n 2) Ä_ nb1 â(2n) ¸ 3n 2 ¸ 1 Ê kxk 23 † #†52†8†4â†6(3n 1)xn ¹ 1 Ê kxk n lim Ä _ 2n 2 Ê the radius of convergence is 23 nb1 (2n1)(2n3)(x1) 82. n lim ¹ 3†5†47†â 9†14â(5n1)(5n4) Ä_ ¸ 2n 3 ¸ 1 Ê kxk 52 † 34††59††714ââ(2n(5n1)x1)n ¹ 1 Ê kxk n lim Ä _ 5n 4 Ê the radius of convergence is 52 n n n kœ2 kœ2 kœ2 83. ! ln ˆ1 k"# ‰ œ ! ln ˆ1 k" ‰ ln ˆ1 k" ‰‘ œ ! cln (k 1) ln k ln (k 1) ln kd œ cln 3 ln 2 ln 1 ln 2d cln 4 ln 3 ln 2 ln 3d cln 5 ln 4 ln 3 ln 4d cln 6 ln 5 ln 4 ln 5d á cln (n 1) ln n ln (n 1) ln nd œ cln 1 ln 2d cln (n 1) ln nd after cancellation n _ kœ2 kœ2 1 ‰ 1‰ Ê ! ln ˆ1 k"# ‰ œ ln ˆ n2n Ê ! ln ˆ1 k"# ‰ œ n lim ln ˆ n 2n œ ln "# is the sum Ä_ n n kœ2 kœ2 84. ! k# " 1 - œ #" ! ˆ k " 1 k " 1 ‰ œ #" ˆ 11 3" ‰ ˆ #" 4" ‰ ˆ 3" 5" ‰ ˆ 4" 6" ‰ á ˆ n " # n" ‰ # 2(n 1) 2n ˆ n " 1 n " 1 ‰‘ œ #" ˆ 1" #" n" n " 1 ‰ œ #" ˆ #3 n" n " 1 ‰ œ #" ’ 3n(n 1)2n(n “ œ 3n4n(n n 1)2 1) _ "ˆ3 Ê ! k#"1 œ n lim 1n n 1 1 ‰ œ 34 Ä_ # 2 kœ2 1)x 85. (a) n lim ¹ 1†4†7â(3n(3n2)(3n 3)! Ä_ 3nb3 (3n ") $ † 1†4†7â(3n)! (3n 2)x3n ¹ 1 Ê kx k n lim Ä _ (3n 1)(3n 2)(3n 3) œ kx$ k † 0 1 Ê the radius of convergence is _ _ _ nœ1 _ nœ1 _ nœ1 nœ2 (3n 2) 3n ! 1†4†7â(3n 2) x3nc1 (b) y œ 1 ! 1†4†7â x Ê dy (3n)! dx œ (3n 1)! Ê d# y ! 1†4†7â(3n 2) x3nc2 œ x ! 1†4†7â(3n5) x3nc2 dx# œ (3n 2)! (3n3)! _ (3n 2) 3n œ x Œ1 ! 1†4†7â x œ xy 0 (3n)! nœ1 Ê a œ 1 and b œ 0 _ 86. (a) x# x# # # # # # $ # $ % & ! (1)n xn which 1 x œ 1 (x) œ x x (x) x (x) x (x) á œ x x x x á œ nœ2 converges absolutely for kxk 1 _ _ nœ2 nœ2 (b) x œ 1 Ê ! (1)n xn œ ! (1)n which diverges _ _ 87. Yes, the series ! an bn converges as we now show. Since ! an converges it follows that an Ä 0 Ê an 1 nœ1 nœ1 _ _ nœ1 nœ1 for n some index N Ê an bn bn for n N Ê ! an bn converges by the Direct Comparison Test with ! bn _ 88. No, the series ! an bn might diverge (as it would if an and bn both equaled n) or it might converge (as it would if nœ1 an and bn both equaled "n ). _ _ nœ1 kœ1 !(xk1 xk ) œ lim (xn1 x" ) œ lim (xn1 ) x" Ê both the series and 89. ! (xn1 xn ) œ n lim Ä_ nÄ_ nÄ_ sequence must either converge or diverge. Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 641 642 Chapter 10 Infinite Sequences and Series 90. It converges by the Limit Comparison Test since n lim Ä_ Š 1 banan ‹ an _ " œ n lim œ 1 because ! an converges Ä _ 1 a n nœ1 and so an Ä 0. _ a" ˆ "# ‰ a# ˆ 3" 4" ‰ a% ˆ 5" 6" 7" 8" ‰ a) 91. ! ann œ a" a## a3$ a4% á nœ1 " " " ‰ ˆ "9 10 11 á 16 a"' á 92. an œ ln"n for n 2 Ê a# a$ " # (a# a% a) a"' á ) which is a divergent series á , and ln"# ln"4 ln"8 á œ ln"# # ln" 2 3 ln" 2 á a% _ œ ln"# ˆ1 #" "3 á ‰ which diverges so that 1 ! n ln" n diverges by the Integral Test. nœ2 CHAPTER 10 ADDITIONAL AND ADVANCED EXERCISES _ 1. converges since (3n #")Ð2n 1ÑÎ2 (3n "2)$Î# and ! (3n "2)$Î# converges by the Limit Comparison Test: nœ1 lim nÄ_ Š " Š $Î# ‹ n (3n " ‹ 2)$Î# ˆ 3n n 2 ‰ œ n lim Ä_ $Î# œ 3$Î# _ 2. converges by the Integral Test: '1 atan" xb $ $ # dx x # 1 " $ " b $ $ 1 œ lim ’ atan 3 xb “ œ lim ’ atan 3 bb 192 “ bÄ_ " bÄ_ $ 1 1 71 œ Š 24 192 ‹ œ 192 c2n e 3. diverges by the nth-Term Test since n lim a œ n lim (1)n tanh n œ lim (1)n Š 11 (1)n ec2n ‹ œ n lim Ä_ n Ä_ Ä_ bÄ_ does not exist 4. converges by the Direct Comparison Test: n! nn Ê ln (n!) n ln (n) Ê lnln(n!) (n) n Ê logn (n!) n Ê lognn $(n!) n"# , which is the nth-term of a convergent p-series 12 1 †2 12 ˆ 42††53 ‰ ˆ 31††42 ‰ 5. converges by the Direct Comparison Test: a" œ 1 œ (1)(3)(2) # , a# œ 3†4 œ (2)(4)(3)# , a$ œ _ "2 12 12 ! ˆ 53††64 ‰ ˆ 42††53 ‰ ˆ 31††42 ‰ œ (4)(6)(5) œ (3)(5)(4) # , a% œ # , á Ê 1 (n 1)(n 3)(n 2)# represents the nœ1 given series and (n 1)(n 123)(n 2)# 12 , which is the nth-term of a convergent p-series n% anb1 n 6. converges by the Ratio Test: n lim œ n lim œ01 Ä _ an Ä _ (n 1)(n 1) 7. diverges by the nth-Term Test since if an Ä L as n Ä _, then L œ 1 " L Ê L# L 1 œ 0 Ê L œ 1 „# _ _ nœ1 nœ1 8. Split the given series into ! 32n"b1 and ! 32n2n ; the first subseries is a convergent geometric series and the n 2n É second converges by the Root Test: n lim 32n œ n lim Ä_ Ä_ n n n È 2È È 9 œ "9†1 œ 9" 1 9. f(x) œ cos x with a œ 13 Ê f ˆ 13 ‰ œ 0.5, f w ˆ 13 ‰ œ #3 , f ww ˆ 13 ‰ œ 0.5, f www ˆ 13 ‰ œ È # È È3 Ð4Ñ ˆ 13 ‰ œ 0.5; # ,f $ cos x œ "# #3 ˆx 13 ‰ 4" ˆx 13 ‰ 1#3 ˆx 13 ‰ á Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. È5 Á0 Chapter 10 Additional and Advanced Exercises 643 10. f(x) œ sin x with a œ 21 Ê f(21) œ 0, f w (21) œ 1, f ww (21) œ 0, f www (21) œ 1, f Ð4Ñ (21) œ 0, f Ð5Ñ (21) œ 1, $ & ( f Ð6Ñ (21) œ 0, f Ð7Ñ (21) œ 1; sin x œ (x 21) (x 3!21) (x 5!21) (x 7!21) á # $ 11. ex œ 1 x x#! x3! á with a œ 0 12. f(x) œ ln x with a œ 1 Ê f(1) œ 0, f w (1) œ 1, f ww (1) œ 1,f www (1) œ 2, f Ð4Ñ (1) œ 6; # $ % ln x œ (x 1) (x # 1) (x 3 1) (x 4 1) á 13. f(x) œ cos x with a œ 221 Ê f(221) œ 1, f w (221) œ 0, f ww (221) œ 1, f www (221) œ 0, f Ð4Ñ (221) œ ", f Ð5Ñ (221) œ 0, f Ð6Ñ (221) œ 1; cos x œ 1 "# (x 221)# 4!" (x 221)% 6!" (x 221)' á 14. f(x) œ tan" x with a œ 1 Ê f(1) œ 14 , f w (1) œ "# , f ww (1) œ "# , f www (1) œ "# ; # $ tan" x œ 14 (x 2 1) (x 4 1) (x 121) á n 15. Yes, the sequence converges: cn œ aan bn b1În Ê cn œ b ˆˆ ba ‰ 1‰ ˆ a ‰n ln ˆ a ‰ œ ln b lim ˆb a ‰n 1b œ ln b nÄ_ b 1În n lnˆˆ ba ‰ 1‰ n nÄ_ Ê n lim c œ ln b lim Ä_ n 0†ln ˆ ba ‰ c œ eln b œ b. 0 1 œ ln b since 0 a b. Thus, n lim Ä_ n _ _ _ 2 16. 1 10 103 # 107 $ 102 % 103 & 107 ' á œ 1 ! 103n2 2 ! 103n3 1 ! 1073n _ _ _ nœ0 nœ0 nœ0 œ 1 ! 103n2 1 ! 103n3 2 ! 103n7 3 œ 1 nœ1 nœ1 2 ‰ ˆ 10 Š 103# ‹ "‰ 1 ˆ 10 $ "‰ 1 ˆ 10 nœ1 Š 107$ ‹ $ "‰ 1 ˆ 10 $ 30 7 999237 412 œ 1 200 œ 333 999 999 999 œ 999 kb1 17. sn œ ! 'k n1 kœ0 Ê sn œ '0 1 dxx# '1 1 dxx# á 'nc1 1 dxx# Ê sn œ '0 1 dxx# 1 dx 1 x# 2 n n Ê n lim s œ n lim atan" n tan" 0b œ 1# Ä_ n Ä_ nb1 # 1) 18. n lim † (n 1)(2x ¹ uunbn 1 ¹ œ n lim ¹ (n 1)x ¹ œ n lim ¹ x † (n 1) ¹ œ ¸ 2x x 1 ¸ 1 nxn Ä_ Ä _ (n 2)(2x 1)n 1 Ä _ 2x 1 n(n 2) n Ê kxk k2x 1k ; if x 0, kxk k2x 1k Ê x 2x 1 Ê x 1; if "# x 0, kxk k2x 1k Ê x 2x 1 Ê 3x 1 Ê x "3 ; if x #" , kxk k2x 1k Ê x 2x 1 Ê x 1. Therefore, the series converges absolutely for x 1 and x "3 . 19. (a) No, the limit does not appear to depend on the value of the constant a (b) Yes, the limit depends on the value of b (c) cos ˆ a ‰ n s œ Š1 n n ‹ œ n lim Ä_ a â‰ â‰ 01 n sin n cos n cos ˆ na ‰ 10 1 n a‰ n ˆ cos n 1 Îb lim Š1 nÄ_ Ê ln s œ ln Œ1 œ bn cos ˆ na ‰ n ˆ "n ‰ Ê n lim ln s œ Ä_ œ 1 Ê n lim sœe Ä_ " 1 " cos ˆ na ‰ Œ n a a a n sin ˆ n ‰ cos ˆ n ‰ n# Š "# ‹ n ¸ 0.3678794412; similarly, ‹ œe n 1În _ an ‰ 20. ! an converges Ê n lim a œ 0; n lim ’ˆ 1 sin “ # Ä_ n Ä_ nœ1 an ‰ ˆ 1sin œ n lim œ # Ä_ Ä_ 1sin Šnlim an ‹ # œ "# Ê the series converges by the nth-Root Test Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 0 œ 1sin # 644 Chapter 10 Infinite Sequences and Series nb1 nb1 21. n lim ¹ uunbn 1 ¹ 1 Ê n lim ¹ b x † ln n ¹ 1 Ê kbxk 1 Ê "b x b" œ 5 Ê b œ „ 5" Ä_ Ä _ ln (n 1) bn xn 22. A polynomial has only a finite number of nonzero terms in its Taylor series, but the functions sin x, ln x and ex have infinitely many nonzero terms in their Taylor expansions. $ $ $ Šax a 3!x á‹ Šx x3! á‹ x $ & sin (ax) sin xx œ lim œ lim ’ a x# 2 a3! 3!" Š a5! 5!" ‹ x# á “ $ $ x x xÄ0 xÄ0 xÄ0 $ xx is finite if a 2 œ 0 Ê a œ 2; lim sin 2x xsin œ 23! 3!" œ 76 $ xÄ0 23. lim # # 24. lim cos#axx# b œ 1 xÄ0 Ê lim Ê b œ 1 and a œ „ 2 25. (a) (b) #x# xÄ0 (n 1)# un œ 1 n2 n"# unb1 œ n# un n1 1 0 unb1 œ n œ 1 n n# % % a x a x Œ1 # 4! á b # xÄ0 _ Ê C œ 2 1 and ! n"# converges nœ1 _ Ê C œ 1 Ÿ 1 and ! "n diverges nœ1 6 26. # # œ 1 Ê lim Š "2x#b a4 a48x á ‹ œ 1 5n# 3 Š4‹ Š#‹ 2n(2n 1) un 4n# 2n 5 unb1 œ (2n 1)# œ 4n# 4n 1 œ 1 n 4n# 4n 1 œ 1 n # Ê C œ 3# 1 and kf(n)k œ 4n# 5n4n 1 œ 5 Ÿ5 Š4 4n "# ‹ _ _ nœ1 nœ1 nœ1 n# after long division _ Ê ! un converges by Raabe's Test n _ – Š4n# c 4n b 1‹ — nœ1 27. (a) ! an œ L Ê an# Ÿ an ! an œ an L Ê ! an# converges by the Direct Comparison Test (b) converges by the Limit Comparison Test: n lim Ä_ an Š1c an ‹ _ an " œ n lim œ 1 since ! an converges and Ä _ 1 an # $ nœ1 therefore x lim a œ0 Ä_ n 28. If 0 an 1 then kln (1 an )k œ ln (1 an ) œ an a#n a3n á an an# an$ á œ 1 anan , a positive term of a convergent series, by the Limit Comparison Test and Exercise 27b _ _ nœ1 nœ1 d 29. (1 x)" œ 1 ! xn where kxk 1 Ê (1"x)# œ dx (1 x)" œ ! nxn1 and when x œ #" we have # $ n 1 4 œ 1 2 ˆ "# ‰ 3 ˆ "# ‰ 4 ˆ "# ‰ á n ˆ "# ‰ á _ _ _ _ nœ1 nœ1 nœ1 # 2xx# ! n(n 1)xn1 œ 2 $ Ê ! n(n 1)xn œ 2x $ 30. (a) ! xn1 œ 1xx Ê ! (n 1)xn œ (1 x)# Ê (1x) (1x) nœ1 Ê _ ! n(n n 1) œ x nœ1 2 x 2x# $ œ (x 1)$ , kxk 1 Š1 "x ‹ _ $ # (b) x œ ! n(nxn ") Ê x œ (x 2x 1)$ Ê x 3x x 1 œ 0 Ê x œ 1 Š1 # nœ1 "Î$ È57 "Î$ È Š1 957 ‹ 9 ‹ ¸ 2.769292, using a CAS or calculator _ 31. (a) " d ˆ " ‰ d # $ # $ ! nxn1 (1x)# œ dx 1x œ dx a1 x x x á b œ 1 2x 3x 4x á œ nœ1 _ (b) 2 n 1 from part (a) we have ! n ˆ 56 ‰ ˆ "6 ‰ œ ˆ 6" ‰ ’ 1 "ˆ 5 ‰ “ œ 6 6 nœ1 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. Chapter 10 Additional and Advanced Exercises _ (c) from part (a) we have ! npn1 q œ (1 q p)# œ qq# œ "q nœ1 _ _ kœ1 kœ1 " _ _ _ kœ1 kœ1 kœ1 ˆ ‰ 32. (a) ! pk œ ! 2k œ 1#ˆ " ‰ œ 1 and E(x) œ ! kpk œ ! k2k œ "# ! k21k œ ˆ "# ‰ # " # œ 2 1ˆ "# ‰‘ by Exercise 31(a) _ _ kœ1 kœ1 _ kc1 k ˆ5‰ _ _ kœ1 kœ1 _ kc1 (b) ! pk œ ! 56k œ 5" ! ˆ 56 ‰ œ ˆ 5" ‰ ’ 1 6ˆ 5 ‰ “ œ 1 and E(x) œ ! kpk œ ! k 56k œ 6" ! k ˆ 56 ‰ kœ1 6 k 1 kœ1 œ ˆ "6 ‰ " # œ 6 1ˆ 56 ‰‘ _ _ _ kœ1 kœ1 (c) ! pk œ ! k(k"1) œ ! ˆ k" k " 1 ‰ œ kœ1 _ œ! kœ1 _ _ lim ˆ1 k " 1 ‰ œ 1 and E(x) œ ! kpk œ ! k Š k(k " 1) ‹ kÄ_ kœ1 kœ1 " k 1 , a divergent series so that E(x) does not exist C! e kt! ˆ1 e nkt! ‰ 1e kt! kt! ! lim R œ 1C! ee kt! œ ektC! 1 nÄ_ n e 1 a1 e n b e " a1 e "! b " (b) Rn œ 1 e " Ê R" œ e ¸ 0.36787944 and R"! œ 1 e " ¸ 0.58195028; 33. (a) Rn œ C! ekt! C! e2kt! á C! enkt! œ Ê Rœ R œ e" 1 ¸ 0.58197671; R R"! ¸ 0.00002643 Ê R RR"! 0.0001 Þ1n e Þ1 ˆ1 e Þ1n ‰ R , # œ #" ˆ eÞ1 " 1 ‰ ¸ 4.7541659; Rn R# Ê 1eÞ1 e 1 ˆ #" ‰ ˆ eÞ1 " 1 ‰ 1 e Þ1 n n 1 enÎ10 "# Ê enÎ10 "# Ê 10 ln ˆ "# ‰ Ê 10 ln ˆ "# ‰ Ê n 6.93 (c) Rn œ Ê Ê nœ7 ! 34. (a) R œ ektC! Ê Rekt! œ R C! œ CH Ê ekt! œ CCHL Ê t! œ k" ln Š CCHL ‹ 1 " (b) t! œ 0.05 ln e œ 20 hrs 2 ‰ (c) Give an initial dose that produces a concentration of 2 mg/ml followed every t! œ 0.0" # ln ˆ 0.5 ¸ 69.31 hrs by a dose that raises the concentration by 1.5 mg/ml " 0.1 ‰ ‰ (d) t! œ 0.2 ln ˆ 0.03 œ 5 ln ˆ 10 3 ¸ 6 hrs Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley. 645 646 Chapter 10 Infinite Sequences and Series NOTES: Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.

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