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

726 Chapter 11 Infinite Sequences and Series
9–26 Determine whether the series is convergent or divergent.
9. ò
n−1
1
n s2
11. 1 1 1 8 1 1 27 1 1 64 1 1
125 1 ∙ ∙ ∙
12.
13.
1
5 1 1 7 1 1 9 1 1 11 1 1 13 1 ∙ ∙ ∙
1
3 1 1 7 1 1 11 1 1 15 1 1 19 1 ∙ ∙ ∙
10. ò n 20.9999
n−3
14. 1 1 1
2s2 1 1
3s3 1 1
4s4 1 1
5s5 1 ∙ ∙ ∙
sn 1 4
15. ò
16. ò
n−1 n 2 n−1
1
17. ò
n−1 n 2 1 4
n
19. ò
3
n−1 n 4 1 4
1
21. ò
n−2 n ln n
23. ò ke 2k
k−1
1
25. ò
n−1 n 2 1 n 3
sn
1 1 n 3y2
1
18. ò
n−1 n 2 1 2n 1 2
3n 2 4
20. ò
n−3 n 2 2 2n
ln n
22. ò
n−2 n 2
24. ò ke 2k 2
k−1
26. n
ò
n−1 n 4 1 1
27–28 Explain why the Integral Test can’t be used to determine
whether the series is convergent.
cos n
27. ò
n−1 sn
cos
28. ò
2 n
n−1 1 1 n 2
29–32 Find the values of p for which the series is convergent.
1
29. ò
n−2 nsln nd p
31. ò ns1 1 n 2 d p
n−1
30. 1
ò
n−3 n ln n flnsln ndg p
ln n
32. ò
n−1 n p
33. The Riemann zeta-function is defined by
1
sxd − ò
n−1 n x
and is used in number theory to study the distribution of
prime numbers. What is the domain of ?
CAS
34. Leonhard Euler was able to calculate the exact sum of the
p-series with p − 2:
1
s2d − ò
n−1 n − 2
2 6
(See page 720.) Use this fact to find the sum of each series.
1
(a) ò
n−2 n 2
1
(c) ò
n−1 s2nd 2
(b) 1
ò
n−3 sn 1 1d 2
35. Euler also found the sum of the p-series with p − 4:
1
s4d − ò
n−1 n − 4
4 90
Use Euler’s result to find the sum of the series.
(a) ò
n−1S
nD
3 4
1
(b) ò
k−5 sk 2 2d 4
36. (a) Find the partial sum s 10 of the series o ǹ−1 1yn4 . Estimate
the error in using s 10 as an approximation to the
sum of the series.
(b) Use (3) with n − 10 to give an improved estimate of the
sum.
(c) Compare your estimate in part (b) with the exact value
given in Exercise 35.
(d) Find a value of n so that s n is within 0.00001 of the sum.
37. (a) Use the sum of the first 10 terms to estimate the sum of
the series oǹ−1 1yn2 . How good is this estimate?
(b) Improve this estimate using (3) with n − 10.
(c) Compare your estimate in part (b) with the exact value
given in Exercise 34.
(d) Find a value of n that will ensure that the error in the
approximation s < s n is less than 0.001.
38. Find the sum of the series oǹ−1 ne22n correct to four decimal
places.
39. Estimate o ǹ−1 s2n 1 1d26 correct to five decimal places.
40. How many terms of the series oǹ−2 1yfnsln nd2 g would you
need to add to find its sum to within 0.01?
41. Show that if we want to approximate the sum of the series
o ǹ−1 n21.001 so that the error is less than 5 in the ninth decimal
place, then we need to add more than 10 11,301 terms!
42. (a) Show that the seriesoǹ−1 sln nd2 yn 2 is convergent.
(b) Find an upper bound for the error in the approximation
s < s n.
(c) What is the smallest value of n such that this upper
bound is less than 0.05?
(d) Find s n for this value of n.
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