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

Section 11.6 Absolute Convergence and the Ratio and Root Tests 743
sin n
5. ò
6. ò s21d n21
n−1 2 n n−1
n
n 2 1 4
7–24 Use the Ratio Test to determine whether the series is
convergent or divergent.
n
7. ò
n−1 5 8. s22d
ò
n
n
n−1 n 2
9. ò s21d n21
n−1
1
11. ò
k−1 k!
10
13. ò
n
n−1 sn 1 1d4 2n11
n
15. ò
n
n−1 s23d n21
cossny3d
17. ò
n−1 n!
n
19. ò
100 100 n
n−1 n!
3 n
2 n n 3 10. ò
n−0
12. ò ke 2k
k−1
s23d n
s2n 1 1d!
14. n!
ò
n−1 100 n
16. n
ò
10
n−1 s210d n11
n!
18. ò
n−1 n n
s2nd!
20. ò
n−1 sn!d 2
21. 1 2 2!
1 ? 3 1 3!
1 ? 3 ? 5 2 4!
1 ? 3 ? 5 ? 7 1 ∙ ∙ ∙
22.
1 s21d n21 n!
1 ? 3 ? 5 ? ∙ ∙ ∙ ? s2n 2 1d 1 ∙ ∙ ∙
2
3 1 2 ? 5
3 ? 5 1 2 ? 5 ? 8
3 ? 5 ? 7 1 2 ? 5 ? 8 ? 11
3 ? 5 ? 7 ? 9 1 ∙ ∙ ∙
2 ? 4 ? 6 ? ∙ ∙ ∙ ? s2nd
23. ò
n−1 n!
2
24. ò
n n!
s21d n
n−1 5 ? 8 ? 11 ? ∙ ∙ ∙ ? s3n 1 2d
25–30 Use the Root Test to determine whether the series is
convergent or divergent.
25.
n−1S ò
n2 n
1 1
s22d
26.
2n 2 1 1D
ò
n
n−1 n n
s21d
27. ò
n21
n−2 sln nd 28. ò
n
n−1S 22n
5n
n 1 1D
29. ò 1
n−1S1
nDn 1 2 30. ò sarctan nd n
n−0
31–38 Use any test to determine whether the series is absolutely
convergent, conditionally convergent, or divergent.
s21d
31. ò
n
32. ò
n−2 ln n
n−1S 1 2 n
n
2 1 3nD
33. ò
n−1
s29d n
n10 n11
35.
n−2S ò
n
37. ò
n−1
n
ln nD
34. n5
ò
2n
n−1 10 n11
sinsny6d
36. ò
n−1 1 1 nsn
s21d n arctan n
s21d
38. ò
n
n 2 n−2 n ln n
39. The terms of a series are defined recursively by the equations
a 1 − 2 a n11 − 5n 1 1
4n 1 3 an
Determine whether o a n converges or diverges.
40. A series o a n is defined by the equations
a 1 − 1
a n11 − 2 1 cos n
sn
Determine whether o a n converges or diverges.
41–42 Let hb nj be a sequence of positive numbers that converges
to 1 2 . Determine whether the given series is absolutely convergent.
n
b n cos n
41. ò
n−1 n
42. ò
n−1
a n
s21d n n!
n n b 1b 2b 3 ∙ ∙ ∙ b n
43. For which of the following series is the Ratio Test inconclusive
(that is, it fails to give a definite answer)?
1
(a) ò
n−1 n 3
s23d
(c) ò
n21
n−1 sn
ò
n−1
(b) n
ò
n−1 2 n
sn
(d) ò
n−1 1 1 n 2
44. For which positive integers k is the following series
convergent?
sn!d 2
sknd!
45. (a) Show that o ǹ−0 x n yn! converges for all x.
(b) Deduce that lim n l ` x n yn! − 0 for all x.
46. Let o a n be a series with positive terms and let r n − a n11ya n.
Suppose that lim n l ` r n − L , 1, so o a n converges by the
Ratio Test. As usual, we let R n be the remainder after n terms,
that is,
R n − a n11 1 a n12 1 a n13 1 ∙ ∙ ∙
(a) If hr nj is a decreasing sequence and r n11 , 1, show, by
summing a geometric series, that
an11
R n <
1 2 r n11
(b) If hr nj is an increasing sequence, show that
an11
R n <
1 2 L
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