Thomas Calculus 13th [Solutions]

722 Chapter 10 Infinite Sequences and Series
15. diverges;
3 3 1
,
n
n 1 n 1
n
which is a divergent p-series with p
1
2
2 1
n n n
n 1 n 1
16. converges; 2 , 3 2
which is a convergent p-series with p
3
2
17. converges; a geometric series with r
1
8
1
18. diverges;
8
8
1
n n
n 1 n 1
and since
1
n 1 n
diverges, 8
1
n
n 1
diverges
19. diverges by the Integral Test:
n
ln x 1 2
ln ln 2
ln x
2 x 2
2 x
dx n dx
20. diverges by the Integral Test:
ln x
;
2
dx ln , dx
, t
t x dt
x x
dx e dt
t 2 t 2 t 2
b
b 2 (ln 2) 2
te dt te e e b e
ln 2 b
ln 2 b
lim 2 4 lim 2 ( 2) 2 (ln 2 2)
21. converges; a geometric series with r
2
3
1
n
22. diverges; lim 5 lim 5 ln 5 lim ln 5 5
0
n
n 4 3 n 4 ln 4 n
n
n
ln 4 4
n
23. diverges;
2 2 1
,
n 1 n 1
n 0 n 0
which diverges by the Integral Test
n dx
1 2x
1 2
24. diverges by the Integral Test:
1
ln(2n 1) as n
25. diverges; lim a lim
2
lim
2 ln 2
0
n
n n
n 1
n
1
n
n
26. diverges by the Integral Test:
1
; ln n 1 ln 2
n u x
dx
n
1 x x 1 du dx 2
x
1
du
u
as n
27. diverges;
1
2 n
1
n
n
n
n
ln n
n n
2
lim lim lim 0
28. diverges; lim a lim 1
1 e 0
n
n
n
n
n
29. diverges; a geometric series with r 1
ln 2
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