Thomas Calculus 13th [Solutions]

730 Chapter 10 Infinite Sequences and Series
10. Compare with
1
, which is a divergent p-series since p
1
1. Both series have positive terms for n 1.
2
n 1
n
2
n 1
2 an
n
2
n n
2
n n 2n
1 2
n
bn
n 1/ n n
2
n 2 n
2
n 2 n
2n
n
2
lim lim lim lim lim lim 1 1 0. Then by Limit
Comparison Test,
2
n 1
n
n
1
2
diverges.
11. Compare with
1,
n
n 2
which is a divergent p-series since p 1 1. Both series have positive terms for n 2.
n( n 1)
an
n
2 1 ( n 1)
n 3
n 2
3n 2
2n 6n
2 6
n
bn
n
1/ n
n
3
n
2
n n 1 n
2
3n 2n
1 n
6n
2
n
6
lim lim lim lim lim lim 1 0. Then by Limit
Comparison Test,
2
n( n 1)
( n 1)( n 1)
n 2
diverges.
12. Compare with
1
, which is a convergent geometric series, since | r |
1
1. Both series have positive
terms for n 1.
n
n
2
1
2
n
an
3 4
n
n
4
n
4 ln 4
n
bn
n
n
1/2 n
n
3 4 n
n
4 ln 4
lim lim lim lim 1 0. Then by Limit Comparison Test,
2
n
n
2
n
3 4
1
converges.
13. Compare with
1
, which is a divergent p-series since p
1
1. Both series have positive terms for n 1.
n
n 1
5
n
a
4
n
n
5 5
n
n
n
n
n
bn
n 1/ n n 4 n
4
lim lim lim lim . Then by Limit Comparison Test,
5
2
n 1
n
n
n 4
diverges.
14. Compare with
2
terms for n 1.
n
,
which is a convergent geometric series since | r |
2
1. Both series have positive
5
5
n 1
2n
3
n
an
5n
4
lim lim lim
10n 15
n
exp lim ln
10n 15
n
exp lim n ln
10n
15
n
n
bn
n (2/5) n
10n 8
n
10n 8
n
10n
8
ln 10n
15 10 10
10n 8
2 2
10n 15 10n 8
70n 70n
1/ 2 (10 15)(10 8)
2
x
n
n 1/ n n
n n
n 100n 230n
120
exp lim exp lim exp lim exp lim
140n
140
200n
230
n
200
7/10
exp lim exp lim e 0. Then by Limit Comparison Test,
2n
3
n
5n
4
n 1
n
converges.
15. Compare with
1,
which is a divergent
n
n 2
1
an
ln n n 1
n
bn
n
1/ n
n
ln n
n
1/ n
n
p -series, since p 1 1. Both series have positive terms for n 2.
lim lim lim lim lim n . Then by Limit Comparison Test,
1
n
ln
2
n
diverges.
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