MATH 152, Fall 2014 Week In Review Week 11 ∞ X 1 1 √ −√ . 1. Determine the sum of the series n n+1 n=1 2. Let an = ∞ X 2n . Determine whether {an } is convergent. Determine whether an is convergent. 3n + 1 n=1 32 8 16 + + + · · · is convergent or divergent. If it is convergent, nd its sum. 5 25 125 ∞ X 1 4. Determine whether the series is convergent or divergent. If it is convergent, nd its sum. e2n n=1 3. Determine whether the series 4 + 5. Determine whether the series ∞ X [2(0.1)n + (0.2)n ] is convergent or divergent. If it is convergent, nd its sum. n=1 6. Determine whether the series 7. Determine whether the series 8. Determine whether the series 9. Determine whether the series ∞ X n2 is convergent or divergent. If it is convergent, nd its sum. 3(n + 1)(n + 2) n=1 ∞ X 1 is convergent or divergent. If it is convergent, nd its sum. n(n + 2) n=1 ∞ X 1 is convergent or divergent. If it is convergent, nd its sum. 2−1 4n n=1 ∞ X √ n=1 10. Determine whether the series n is convergent or divergent. If it is convergent, nd its sum. 1 + n2 ∞ X 1 is convergent or divergent. If it is convergent, nd its sum. 1 + 2−n n=1 11. Find the values for x such that the series ∞ X (x − 3)n is convergent. Find the sum for those values of x. n=1 ∞ X 1 12. Find the values for x such that the series is convergent. Find the sum for those values of x. n x n=1 13. Find the values for x such that the series ∞ X tann x is convergent. Find the sum for those values of x. n=1 ∞ X 14. Find the sum of the series 1 . (4n + 1)(4n − 3) n=1 15. Find the sum of the series ∞ X n2 + 3n + 1 . (n2 + n)2 n=1 16. If the n − th partial sum of a series ∞ X an is n=1 sn = nd an and ∞ X n=1 an . n−1 n+1 17. If the n − th partial sum of a series ∞ X an is n=1 sn = 3 − n2−n nd an and ∞ X an . n=1 18. What is the value for c if ∞ X (1 + c)−n such that the series is convergent.? n=2 19. Suppose that ∞ X an an 6= 0, is known to be convergent. Prove that n=1 ∞ X 1 is divergent. a n=1 n 20. Determine whether the following series are absolutely convergent. (a) ∞ X (−3)n . n3 n=1 (b) ∞ X (−3)n . n! n=1 ∞ X (−1)n+1 . 2n + 1 n=1 √ ∞ X (−1)n−1 n (d) . n+1 n=1 (c) (e) ∞ X (−1)n+1 5n+1 . (n + 1)2 4n+2 n=1 (f) ∞ X cos(nπ/6) √ . n n n=1 21. For which of the following series is the ratio test inconclusive ? (a) ∞ X 1 . 3 n n=1 ∞ X (−3)n−1 √ . n n=1 √ ∞ X n (c) . 1 + n2 n=1 (b) 22. For which positive integers k is the series ∞ X (n!)2 (kn)! n=1 convergent ? 23. (a) Show that ∞ X xn is convergent for all x. n! n=0 (b) Deduce that limn→∞ 24. Suppose that the series xn =0 n! ∞ X n=0 cn x n is convergent when x = 4 and diverges when x = 6. Is the series convergent ? What can be said about the convergence or divergence of the following series ? ∞ X n=0 cn (a) (b) (c) ∞ X cn 8n . n=0 ∞ X cn (−3)n . n=0 ∞ X (−1)n cn 9n . n=0 25. If ∞ X cn 4n is convergent , does it follow that the following series are convergent? n=0 (a) (b) (c) ∞ X cn (−2)n . n=0 ∞ X n=0 ∞ X cn (3)n . cn (−4)n . n=0 26. Find the radius and interval of convergence of the series . (a) ∞ X xn . n2 n=1 (b) ∞ X (−1)n xn . n2n n=1 (c) ∞ X n (2x − 1)n . n 4 n=0 (d) (e) ∞ X (−1)n n=1 ∞ X (x − 1)n √ . n (x − 4)n . n5n n=1 (f) ∞ X 2n (x − 3)n . n+3 n=0 (g) ∞ X (x + 1)n . n(n + 1) n=1 (h) (i) ∞ X n!(2x − 1)n . n=1 ∞ X nxn . 1 · 3 · 5 · · · · · (2n − 1) n=1 27. If k is a positive integer, nd the radius of convergence of the series . ∞ X (n!)k n x (nk)! n=0 28. A function f is dened by: f (x) = 1 + 2x + x2 + 2x3 + x4 + 2x5 + x6 + · · · That is, its coecients are c2n = 1 and c2n+1 = 2 for all n ≥ 0. Find the interval of convergence of the series and nd an explicit formula for f (x) . 29. If f (x) = f (x). ∞ X cn xn where cn+4 = cn for all n ≥ 0, nd the interval of convergence of the series and a formula for n=0 30. Find a power series representation for the following functions, and determine the interval of convergence. 1 . 1 + 4x2 1 (b) f (x) = 4 . x + 16 1 + x2 . (c) f (x) = 1 − x2 (a) f (x) =