Thomas Calculus 13th [Solutions]

716 Chapter 10 Infinite Sequences and Series
47. s 1 1 1 1 1 1 1 1 1 1 1 1
k ln 3 ln 2 ln 4 ln 3 ln 5 ln 4 ln ( k 1) ln k ln ( k 2) ln ( k 1) ln 2 ln ( k 2)
lim
k
s k
1
ln 2
48.
1 1 1 1 1 1 1 1
sk
tan (1) tan (2) tan (2) tan (3) tan ( k 1) tan ( k) tan ( k) tan ( k 1)
1 1 1
tan (1) tan ( k 1) lim sk
tan (1)
k
2 4 2 4
49. convergent geometric series with sum
1
1
1
2
2
2 1
2 2
50. divergent geometric series with r 2 1
51. convergent geometric series with sum
1
3
2
1
2
1
52.
n 1
lim ( 1) n 0 diverges
n
n
53. The sequence a n cos starting with n 0 is 1, 0, 1, 0,1, 0, 1, 0, . . . , so the sequence of partial
2
sums for the given series is 1,1, 0, 0,1,1, 0, 0, . . . and thus the series diverges.
54. cos ( ) ( 1) n n convergent geometric series with sum
1
1
1
5
5
6
55. convergent geometric series with sum
1
1
1
e
2
2
e
2
e 1
56. lim ln
1
0
n
3
n
diverges
57. convergent geometric series with sum
1
2
2
20 18 2
9 9 9
1
10
58. convergent geometric series with sum
1
1
1
x
x
x
1
59. difference of two geometric series with sum
1 1
1 1
2 1
3 3
3
3 3
2 2
60.
1 n
n
1
n
n
n
lim 1 lim 1 e 0 diverges
n
1
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