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

732 Chapter 11 Infinite Sequences and Series
38. For what values of p does the series oǹ−2 1y sn p ln nd
converge?
39. Prove that if a n > 0 and o a n converges, then o a n
2
also
converges.
40. (a) Suppose that o a n and o b n are series with positive terms
and o b n is convergent. Prove that if
a n
lim − 0
n l ` b n
then o a n is also convergent.
(b) Use part (a) to show that the series converges.
ln n
(i) ò
n−1 n (ii) ò
3
n−1
ln n
sn e n
41. (a) Suppose that o a n and o b n are series with positive terms
and o b n is divergent. Prove that if
a n
lim − `
n l ` b n
then o a n is also divergent.
(b) Use part (a) to show that the series diverges.
1
(i) ò
n−2 ln n
ln n
(ii) ò
n−1 n
42. Give an example of a pair of series o a n and o b n with positive
terms where lim n l ` sa nyb nd − 0 and o b n diverges, but o a n
converges. (Compare with Exercise 40.)
43. Show that if a n . 0 and lim n l ` na n ± 0, then o a n is
divergent.
44. Show that if a n . 0 and o a n is convergent, then o lns1 1 a nd
is convergent.
45. If o a n is a convergent series with positive terms, is it true that
o sinsa nd is also convergent?
46. If o a n and o b n are both convergent series with positive terms,
is it true that o a nb n is also convergent?
The convergence tests that we have looked at so far apply only to series with positive
terms. In this section and the next we learn how to deal with series whose terms are not
necessarily positive. Of particular importance are alternating series, whose terms alternate
in sign.
An alternating series is a series whose terms are alternately positive and negative.
Here are two examples:
1 2 1 2 1 1 3 2 1 4 1 1 5 2 1 6 1 ∙ ∙ ∙ − ò s21d 1 n21
n−1 n
2 1 2 1 2 3 2 3 4 1 4 5 2 5 6 1 6 7 2 ∙ ∙ ∙ − ò s21d n
n−1
n
n 1 1
We see from these examples that the nth term of an alternating series is of the form
a n − s21d n21 b n or a n − s21d n b n
where b n is a positive number. (In fact, b n − | a n | .)
The following test says that if the terms of an alternating series decrease toward 0 in
absolute value, then the series converges.
Alternating Series Test If the alternating series
satisfies
ò s21d n21 b n − b 1 2 b 2 1 b 3 2 b 4 1 b 5 2 b 6 1 ∙ ∙ ∙ b n . 0
n−1
then the series is convergent.
(i) b n11 < b n
(ii) lim
n l `
b n − 0
for all 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.