Covers: 11.2, 11.3, 11.4, 11.5, 11.6, 11.7, 11.8
EXAM 4 (of 4)
Good luck. You must show work to get points.
Name______________________________
#1(27) Determine if each series is C or D. Give the name of the test used ("Def'n", "Div. Test",
"Geom. Series", "p-series", "Int. Test", “CT”, “LCT”, “Ratio Test”, “Root Test”, “AST”, “Alt. pseries”, or “ACT”), your conclusion (C or D) and work for the test.

a)(9)

(1) n
n2
1
ln( n )
Test used______________ Conclusion (C or D) _______
Work:

b)(9)
n!
(2n  1)!
n 1

Test used______________ Conclusion (C or D) _______
Work:
c)(9)

tan 1 (n )
n 1
n2

Test used______________ Conclusion (C or D) _______
Work:
1
#2(10) Circle the correct answer for each series. You do not have to show work for #2.
____a) AC, CC or D.

(1) n
n 1
n2


____b) AC, CC or D.
(1) n
 n
n 1

____c) AC, CC or D.
(1) n

n
n 1

____d) AC, CC or D.

(1) n n
n 1

____e) AC, CC or D.

1
n 1 n
2

____#3(4)

T or F. If an > 0 for all n, then
n 1
a n can NOT be conditionally convergent.
#4(5) Consider approximating the sum S of


(1) n 1
2
=1–
1
2
+
1
2
–
1
2
+
1
2
– … by its nth partial sum sn. What is
3
5
n
2
4
the smallest value of n for which the Alternating Series Estimation Theorem guarantees
that the error in sn (as an approximation to S) is at most 0.1?
n 1
Final Answer. n = _______
____#5(4)
T or F. The Alternating Series Test can NEVER be used to determine that a series
diverges.
2
#6(12) Find the interval of convergence of the power series

( x  3) n
n 1
2n n 2

SHOW WORK.
Final answer: I = _________ __________ , __________
 ( or [
 a # or –  a # or +
__________
 ) or ]
#7(9) Give (written out-you do not have to put in sigma notation) the integral of the series
f(x) = 5 + 10 x + 15 x2 + 20 x3 + 25 x4 + . . .
Answer(6):
 f (x ) dx =
C + _________________________________ + . . .
(Include at least 3 nonzero terms.)
True or False(3). (Circle one.) If R = 1 for f(x), then we must have R > 1 for  f ( x ) dx .
#8(5) Give the Maclaurin Series in sigma notation and intervals of convergence for:
cos(x) =


n 0
I=
3
#9(14)
Give the Taylor series generated by f(x) = ex at x = 5 (i.e., center a = 5).
SHOW WORK.

Answer: f(x) = 
n 0
#10(10)
(x-5)
Give the Maclaurin polynomial of degree 2 generated by f(x) =
SHOW WORK.
Answer: P2(x) = _______________________________________________
4
1
.
x 1