James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

736 Chapter 11 Infinite Sequences and Series
Note The rule that the error (in using s n to approximate s) is smaller than the first
neglected term is, in general, valid only for alternating series that satisfy the conditions
of the Alternating Series Estimation Theorem. The rule does not apply to other types of
series.
1. (a) What is an alternating series?
(b) Under what conditions does an alternating series
converge?
(c) If these conditions are satisfied, what can you say
about the remainder after n terms?
2–20 Test the series for convergence or divergence.
2. 2 3 2 2 5 1 2 7 2 2 9 1 2 11 2 ∙ ∙ ∙
3. 2 2 5 1 4 6 2 6 7 1 8 8 2 10 9 1 ∙ ∙ ∙
4. 1
ln 3 2 1
ln 4 1 1
ln 5 2 1
ln 6 1 1
ln 7 2 ∙ ∙ ∙
s21d
5. ò
n21
n−1 3 1 5n
7. ò s21d 3n 2 1
n
n−1 2n 1 1
9. ò s21d n e 2n
n−1
n
11. ò
2
s21d n11
n−1 n 3 1 4
13. ò s21d n21 e 2yn
n−1
s21d
6. ò
n11
n−0 sn 1 1
n
8. ò
2
s21d n
n−1 n 2 1 n 1 1
sn
10. ò s21d n
n−1 2n 1 3
12. ò s21d n11 ne 2n
n−1
14. ò s21d n21 arctan n
n−1
sinsn 1
15. ò
1 2d
n cos n
16. ò
n−0 1 1 sn n−1 2 n
17. ò
n−1
s21d n sinS nD
19. ò s21d n n
n
n−1 n!
18. ò s21d cosS n
n−1
nD
20. ò s21d n (sn 1 1 2 sn )
n−1
Series Estimation Theorem to estimate the sum correct to four
decimal places.
s20.8d
21. ò
n
n−1 n!
n
22. ò s21d n21
n−1 8 n
23–26 Show that the series is convergent. How many terms
of the series do we need to add in order to find the sum to the
indicated accuracy?
s21d
23. ò
n11
s| error | , 0.00005d
n−1 n 6
(2
24. ò
1 3) n
n−1 n
(| error | , 0.0005)
s21d
25. ò
n21
(| error | , 0.0005)
n−1 n 2 2 n
26. ò
n−1S2
nD
1 n
(| error | , 0.00005)
27–30 Approximate the sum of the series correct to four decimal
places.
s21d
27. ò
n
n−1 s2nd!
29. ò s21d n ne 22n
n−1
s21d
28. ò
n11
n−1 n 6
s21d
30. ò
n21
n−1 n 4 n
31. Is the 50th partial sum s 50 of the alternating series
o ǹ−1 s21dn21 yn an overestimate or an underestimate of the
total sum? Explain.
32–34 For what values of p is each series convergent?
s21d
32. ò
n21
n−1 n p
; 21–22 Graph both the sequence of terms and the sequence of
partial sums on the same screen. Use the graph to make a rough
estimate of the sum of the series. Then use the Alternating
s21d
33. ò
n
n−1 n 1 p
p
n21
sln nd
34. ò s21d
n−2 n
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.