Thomas Calculus 13th [Solutions]

Section 10.5 Absolute Convergence; The Ratio and Root Tests 739
7.
2
n 1 ( n 1) ( n 1) 2 !
( n 1)!3
2( n 1) n
2
n n n
2n
3
n
2
n n
n n
2
( n 2)!
2n
2 2 3 2
( 1)
( n 1) n!3 3 n ( n 2)! 9n 9n
n!3
2n
lim
( 1)
lim
( 1) ( 3)( 2)! !3 lim 5 7 3
n n n
lim
2
2
3 15 7 lim 6 15 lim 6 1 1 ( 1) n n n
2 2
n
n n n ( 2)!
n 27n 18 n n
54 n 18
n
54 9
n!3
n 1
converges
8.
( n 1) 5
n 1
2( n 1) 3 ln ( n 1) 1
n 5
n
(2n
3)ln( n 1)
n
( n 1) 5 5 (2n 3)ln( n 1) 5( n 1) (2n 3) ln( n 1)
lim lim lim
(2 5)ln( 2) n
n n
n n n 5 n
n(2n 5) ln( n 2)
2
1
ln( n 1)
n 1
2 1
n n
n 2
lim 10n 25n 15 lim lim 20n 25 lim lim 20 lim n 2
n 2 5 n
ln( n 2)
n
4n 5
n n
4
n
n 1
5 lim 1 5 1 5 1
n 5
n
1 (2n
3)ln( n 1)
n 2
n
diverges
9.
7 7
lim n
7
n
lim 0 1
n (2n
5) n
2n
5
(2n
5)
n 1
n
n
converges
n
10. lim n 4 4 4
n
lim 0 1
n (3 n) n
3n
(3 n)
n 1
n
n
converges
11. 4 3
n
lim n n lim 4n 3 lim 4 4 1 4n
3
n
3n 5
n
3n 5
n
3 3 3n
5
n 1
n
diverges
12.
1 1 1/
2 1 2 1
2
lim ln n
n
n e lim ln ln 2 1
n
n
e n
n
e
n
1
ln
n 1
2
e 1 diverges
n
n
8
13. lim n
8 1
8
1
2n
lim
1
2
1
9
1
2n
n 3 n 3 3
n n n
n 1
converges
n
14. lim n sin 1 lim sin 1 sin(0) 0 1 sin 1
n
n n n n
n 1
n
converges
15.
2 2
1 1 1
lim n
n
( 1) 1 n lim 1 n e 1 1 1
n converges
n
n
n
n n
n 1
16. ( 1) n n 1 n
lim n
1 1
1 n
lim
1/ n 1
lim 0 1
n
1 n
n n n n n n n
n
n 2
converges
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