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

716 Chapter 11 Infinite Sequences and Series
1
13. ò
n−1 n 2 1 1
14. ò
n−1Ssin 1 n 2 sin 1
n 1 1D
15. Let a n − 2n
3n 1 1 .
(a) Determine whether ha nj is convergent.
(b) Determine whether o ǹ−1 a n is convergent.
16. (a) Explain the difference between
o n
a i and o n
a j
i−1
j−1
(b) Explain the difference between
o n
a i and o n
a j
i−1
i−1
17–26 Determine whether the geometric series is convergent or
divergent. If it is convergent, find its sum.
17. 3 2 4 1 16 3 2 64 9 1 ∙ ∙ ∙ 18. 4 1 3 1 9 4 1 27
16 1 ∙ ∙ ∙
19. 10 2 2 1 0.4 2 0.08 1 ∙ ∙ ∙
20. 2 1 0.5 1 0.125 1 0.03125 1 ∙ ∙ ∙
21. ò 12s0.73d n21
n−1
5
22. ò
n−1 n
s23d
23. ò
n21
24.
n−1 4
ò
n
n−0
3 n11
s22d n
e
25. ò
2n
n−1 6 26. 6 ? 2
ò
2n21
n21
n−1 3 n
27–42 Determine whether the series is convergent or divergent. If
it is convergent, find its sum.
1
27.
3 1 1 6 1 1 9 1 1 12 1 1 15 1 ∙ ∙ ∙
1
28.
3 1 2 9 1 1 27 1 2 81 1 1
243 1 2
729 1 ∙ ∙ ∙
2 1 n
29. ò
n−1 1 2 2n
k
30. ò
2
k−1 k 2 2 2k 1 5
31. ò 3 n11 4 2n 32. ò fs20.2d n 1 s0.6d n21 g
n−1
n−1
1
33. ò
n−1 4 1 e 34. 2
ò
n 1 4 n
2n
n−1 e n
35. ò ssin 100d k
k−1
37. ò
n−1
36. ò
n−1
1
1 1 ( 2 3) n
lnS n2 1 1
38.
2n 2 1 1D ò (s2 ) 2k
k−0
39. ò arctan n
n−1
41. ò
n−1S 1 1
e n
1
nsn 1 1dD
40. ò
n−1S 3 5 nD 1 2 n
e
42. ò
n
n−1 n 2
43–48 Determine whether the series is convergent or divergent
by expressing s n as a telescoping sum (as in Ex am ple 8). If it is
convergent, find its sum.
2
43. ò
n−2 n 2 2 1
3
45. ò
n−1 nsn 1 3d
47. ò se 1yn 2 e 1ysn11d d
n−1
n
44. ò ln
n−1 n 1 1
46. ò
n−4S 1
sn 2 1
1
48. ò
n−2 n 3 2 n
sn 1 1D
49. Let x − 0.99999 . . . .
(a) Do you think that x , 1 or x − 1?
(b) Sum a geometric series to find the value of x.
(c) How many decimal representations does the number 1
have?
(d) Which numbers have more than one decimal
representation?
50. A sequence of terms is defined by
a 1 − 1
Calculate o ǹ−1 an.
a n − s5 2 nda n21
51–56 Express the number as a ratio of integers.
51. 0.8 − 0.8888 . . . 52. 0.46 − 0.46464646 . . .
53. 2.516 − 2.516516516 . . .
54. 10.135 − 10.135353535 . . .
55. 1.234567 56. 5.71358
57–63 Find the values of x for which the series converges. Find
the sum of the series for those values of x.
57. ò s25d n x n
n−1
58. ò sx 1 2d n
n−1
sx 2 2d
59. ò
n
60. ò s24d n sx 2 5d n
n−0 3 n n−0
2
61. ò
n
n−0 x n
63. ò e nx
n−0
62. sin
ò
n x
n−0 3 n
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