Thomas Calculus 13th [Solutions]

704 Chapter 10 Infinite Sequences and Series
56.
lim
n 2 n
2
n lim n
2
1 1 converges (Theorem 5, #2)
n
n
57.
1 n
lim 1 n lim 3
3 n 1
n
1
lim 1
1
n
n
n
n
converges (Theorem 5, #3 and #2)
58.
lim ( n
1 ( n
4)
4)
lim
1 x
x 1 converges; (let x n 4, then use Theorem 5, #2)
n
x
lim ln n
ln n n
59. lim
1 n
1 n
n n lim n
n
1
diverges (Theorem 5, #2)
60.
lim ln n ln ( n 1) lim ln n ln lim n ln 1 0
n n
n 1
n
n 1
converges
61. lim n n
4 n
n
lim 4 n 4 1 4 converges (Theorem 5, #2)
n
n
62.
lim
n 2n 3
1 2 (1 n) lim 3
2
lim 3
1 n
3 9 1 9
n n n
converges (Theorem 5, #3)
63.
! 1 2 3 ( 1)( )
lim n
n n
lim lim 1 0
n
n n n
n n n n n
n
n
and n! 0 lim n!
0
n
n
n n n
converges
64.
n
lim ( 4)
n!
0
n
converges (Theorem 5, #6)
65. lim n! lim 1
6n
n 10 n
6
10
n!
n
diverges (Theorem 5, #6)
66.
lim
n
n! lim 1
n n
2 3 n
6
n
n!
diverges (Theorem 5, #6)
67.
1 (ln )
1 1 1
ln 1 ln 1
lim n
n
lim exp ln lim exp e converges
n
n
n
ln n n
n
ln n
n
n
68. lim ln 1 1 ln lim 1 1 ln e 1 converges (Theorem 5, #5)
n
n
n
n
69.
1
n
3 3
3 1 3 1
3 1 3 1
ln (3 1) ln (3 1)
lim n
n n n
n n
lim exp n ln n lim exp lim exp
n
3n
1
n
3n
1
n n
2
6 6 2 3
lim exp n exp e converges
n
(3n
1)(3n
1) 9
1
n
2
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