Thomas Calculus 13th [Solutions]

732 Chapter 10 Infinite Sequences and Series
25. converges by the Direct Comparison Test; n
n n
n 1
,
3n
1 3n
3
series
the nth term of a convergent geometric
26. converges by the Limit Comparison Test (part 1) with 1 ,
3/2
n
1
n
3/2
3
3 3
lim lim n 2 lim 1 2 1
n 1 n n n n
3 2
n
the nth term of a convergent p-series
27. diverges by the Direct Comparison Test; n ln n ln n ln ln n 1 1 1 and 1
n ln n ln (ln n)
n 3 n
diverges
28. converges by the Limit Comparison Test (part 2) when compared with 1 ,
2
n
n 1
(ln n)
2
n
3
2
(ln n)
2(ln n)
lim lim lim 2 lim ln n 0
n n
n
n
1
n
n
1
n
2
1
n
a convergent p-series
29. diverges by the Limit Comparison Test (part 3) with 1 ,
n
1 1
n ln n n
2 n
1 n
1
n
n
lim lim ln lim lim n
n n n n
2
the nth term of the divergent harmonic series:
30. converges by the Limit Comparison Test (part 2) with 1 ,
5/4
n
the nth term of a convergent p-series
(ln n)
2
3/2
2
2ln n
1
n (ln n) n ln n
n
1
1
1/4
n 1
1/4
n 1
1/4
n
n
5/4
4n 3/4
4n
3/4
lim lim lim 8 lim 8 lim 32 lim 32 0 0
n n n n n n
31. diverges by the Limit Comparison Test (part 3) with 1 ,
n
1
1 ln n
n
1
1 1 ln n
1
n
n
lim lim lim lim n
n n n n
the nth term of the divergent harmonic series:
32. diverges by the Integral Test:
ln( x 1) 1 2 1 2 2
dx u du
2 1 ln 3 lim u b
2 lim b
x
ln 3 2
ln 3
b
b
33. converges by the Direct Comparison Test with 1 ,
3/2
n
2
the nth term of a convergent p-series n
2 2 3 2 3/2
n 2 n n 1 n n n 1 n 1 1 or use Limit Comparison Test with 1 .
3/2
n
2
2
n n 1
n
1
n for
34. converges by the Direct Comparison Test with 1 , the nth term of a convergent p-series
3/2
n
2 2 2 3/2
2
1 3/2 n
n 1 n n 1 n n n n 1 or use Limit Comparison Test with 1 .
2 3/2
3/2
n n 1 n
n
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