Thomas Calculus 13th [Solutions]

734 Chapter 10 Infinite Sequences and Series
44. converges by Limit Comparison Test: compare with 1 , which is a convergent p-series
3
n
n 1
( n 1)!
3
( 2)!
2
3 2
lim
n lim n ( n 1)! 2 2
1/ ( 2)( 1) ( 1)! lim n lim n
n n n n 3 2 2n
3 lim 2
1 0
n n n n n n n n
45. diverges by the Limit Comparison Test (part 1) with 1 ,
n
sin
1
n
lim lim sin x 1
1
n
x 0
x
n
the nth term of the divergent harmonic series:
46. diverges by the Limit Comparison Test (part 1) with 1 ,
n
1
sin
1
n
n
1 1 1
n
n n
lim tan
lim 1 lim 1 sin x
cos 0
cos x x
1 1 1
n n x
the nth term of the divergent harmonic series:
47. converges by the Direct Comparison Test:
p-series and a nonzero constant
tan
n
1
n 2
1.1 1.1
n
2
and 1
n 2 n
n 1 n 1
1.1 1.1
is the product of a convergent
48. converges by the Direct Comparison Test: sec
a convergent p-series and a nonzero constant
1
n 2
1.3 1.3
1
n sec and
2 n n
n
1
n
1
2
n
n
1
2
1.3 1.3
is the product of
49. converges by the Limit Comparison Test (part 1) with 1 :
2
n
2n
2n
lim 1 e 1
n 1 e
coth n
n
2
1
n
2
lim lim coth n lim
n n n
n
e
e
n
e
e
n
n
50. converges by the Limit Comparison Test (part 1) with 1 :
2
n
2n
2n
lim 1 e 1
n 1 e
tanh n
n
2
1
n
2
lim lim tanh n lim
n n n
n
e
e
n
e
e
n
n
51. diverges by the Limit Comparison Test (part 1) with
1
nn
n
1
n
1 : lim lim 1 1
n
n
n
n n
52. converges by the Limit Comparison Test (part 1) with
n n
n
2
1 n
: lim lim n 1
2
n n n
1
n
2
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