Thomas Calculus 13th [Solutions]

Section 10.5 Absolute Convergence; The Ratio and Root Tests 741
29. diverges by the Direct Comparison Test: 1 1 n 1 1 1
n 2 2
n n 2 n
(part 1) with 1 .
n
for n 2 or by the Limit Comparison Test
n
n
1/ n
30. converges by the nth-Root Test: lim n a lim n 1 1 1 1 1 1
n 2
lim
2
lim
2
0 1
n n
n n n
n n n
n n
31. diverges by the nth-Term Test: Any exponenetial with base > 1 grows faster than any fixed power, so
lim an
0.
n
1
32. converges by the Ratio Test: lim a lim ( 1)ln( 1) n
n
n n 2 1
1
2 ln( ) 2
1
n
n
an
n
n n
an
1 ( n 2)( n 3)
33. converges by the Ratio Test: lim lim n!
0 1
n
an
n
( n 1)! ( n 1)( n 2)
34. converges by the Ratio Test:
3
a 1 ( 1) n
n
n
lim lim e 1
n 1 3
1
an
e
n
n
e
n
1
35. converges by the Ratio Test: . lim a lim ( 4)! n
n
n 3! n!3 4 1
1
3!( 1)!3 ( 3)! lim n
3( 1) 3
1
n
n
an
n n n
n
n
36. converges by the Ratio Test:
lim 1
1
lim n
a ( 1)2 ( 2)! n
n
n n 3 n! lim n 1 2 n 2 2
a 1
3 1 3
1
n
n
n
n n
n n 3 ( n 1)! n2 ( n 1)! n
an
1 ( n 1)! (2n 1)!
37. converges by the Ratio Test: lim lim lim n 1 0 1
n
an
n
(2n 3)! n! n
(2n 3)(2n
2)
1
38. converges by the Ratio Test: lim a lim ( 1)! n
n
n n 1 1 1
1
( 1) ! lim n
n
1
lim lim 1
n
n 1
n
1
1
n
n
an
n n n
n
n
n n
e
n
n
39. converges by the Root Test:
lim lim lim n lim n
n
n n
a
n
n n
(ln ) ln lim ln
0 1
n
n n n n
n n
n
n
40. converges by the Root Test: lim lim lim n lim n
n a n
n n
n n
0 1 lim n
n/2
n 1
n n (ln n)
n ln n lim lnn
n
n
n
which is the nth-
41. converges by the Direct Comparison Test: n!ln n ln n n
1 1
n( n 2)! n( n 1)( n 2) n( n 1)( n 2) ( n 1)( n 2) 2
n
term of a convergent p-series
1 3 3
1
42. diverges by the Ratio Test: lim a lim 3 n
n
n
n 2 lim n 3 3
3 1 3
( 1) 2 3 ( 1) 2 2
1
n n
n
an
n n n n
Copyright
2014 Pearson Education, Inc.