Thomas Calculus 13th [Solutions]

Section 10.5 Absolute Convergence; The Ratio and Root Tests 743
2 2
3
6 6
4
24
54. converges by the Direct Comparison Test: a 1 1 1 1 1 1
1 , a
2 2 , a
2 3 , a
2 2 4
,
2 2
1
n!
1
n
a n
which is the nth-term of a convergent geometric series
2 2
55. converges by the Ratio Test:
n 1
lim an
1 lim 2 ( n 1)!( n 1)! (2 n)! 2( 1)( 1) 1 1
(2 2)! lim n n
2 ! ! (2 2)(2 1) lim n
2 1 2
1
n
n
an
n
n n n n
n n
n
n
a 1 (3 3)! !( 1)!( 2)! (3 3)(3 2)(3 1)
56. diverges by the Ratio Test: lim n
n n n n n n
lim lim
n
an
n
( n 1)!( n 2)!( n 3)! (3 n)! n
( n 1)( n 2)( n 3)
lim 3 3n
2 3n
1 3 3 3 27 1
n
n 2 n 3
!
57. diverges by the Root Test: lim n
n
a lim !
2
lim n
n n
2
1
n n n
n n n
n
n
n
( 1) n! n! 58. converges by the Root Test: lim n ! 1 2 3 1
1
2
lim
n
lim n
n
lim n n
n lim
n n n n n n n
n n n n n
n
n
n
n
0 1
n
59. converges by the Root Test: lim n a lim n lim n
1
n n
lim 0 1
n
2
n
n
n n 2 n 2 n 2 ln 2
n
60. diverges by the Root Test: lim n a lim n
2
lim n
n n
1
n n n 4
2 n
n
1
61. converges by the Ratio Test: lim a lim 13 (2 1)(2 1) n n
n
n n 4 2 n! 2 1 1
1 1
4 2 ( 1)! 13 (2 1) lim n
(4 2)( 1) 4
1
n n
n
an
n n
n
n
n
13 (2n 1) 1 2 3 4 (2n 1)(2 n) (2 n)!
62. converges by the Ratio Test: an
2
2
(2 4 2 n) 3 n 1 (2 4 2 n) 3 n 1 2 n n! 3 n 1
n 2
n n n n
n 2
n
lim (2n 2)! 2 ! 3 1 (2 1)(2 2) 3 1 1 3
4 6 2 1 1
1
2
1
2 (2 )! lim lim n n
1 2 2
2 ( 1) 3 1 4 8 4 3 3 3 3
1
n
n
n n
n
n
2 ( n 1)! 3 1
n n
n n n
1
63. Ratio: lim a
lim p
n
1 n
( 1) 1 lim n
p
1
1 p
1
p
no conclusion
n
an
n n
n
n
Root: lim n a lim 1 lim 1 1
n
n
p p p
1 no conclusion
n n n n n
n (1)
p
p
64. Ratio:
n n n n n
conclusion
lim 1
1
lim p
an
1 (ln n) ln 1
1 lim n
ln( 1)
lim n
lim n (1) p
a 1
p
n
1
ln( n 1)
n n
n 1
p
no
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