Thomas Calculus 13th [Solutions]

758 Chapter 10 Infinite Sequences and Series
30.
u n 1
n 1 x n ln ( n) ln ( n)
x n
x x
( 1)ln ( 1) n
n
un
n
n n x
n
n 1
n
ln ( n 1)
lim 1 lim 1 lim lim 1 (1)(1) 1 1
1 x 1; when x 1 we have
n
( 1)
,
n ln n
n 2
which diverges by Exercise 56(a) Section 10.3
(a) the radius is 1; the interval of convergence is 1 x 1
(b) the interval of absolute convergence is 1 x 1
(c) the series converges conditionally at x 1
a convergent alternating series; when x 1 we have
1
n
ln
2 n n
31.
lim 2 3
u 3/2
n 1
1 lim (4x 5) n
n 1 (4 5) 2 lim 2
x n
3/2 2 1
1
1 (4 x 5) 1 4 x
u
5 1
n
n
n
n n ( n 1) (4x
5)
n
3/2
3
2
1 4x 5 1 1 x ; when x 1 we have
when x 3
we have
2
(1)
n 1
n
2n
1
3/2
,
a convergent p-series
n
2n
1
( 1) 1
3/2 3/2
n
n
1 n 1
which is absolutely convergent;
(a) the radius is 1 4 ; the interval of convergence is 1 x 3
2
(b) the interval of absolute convergence is 1 x 3
2
(c) there are no values for which the series converges conditionally
32.
lim 2
un
1
1 lim (3x 1) n
2n 2 2 2
2 4 1 3 x 1 lim n
1
2 4
1 3 x 1 1 1 3 x
u 1 1
n
n
n n
n n (3x
1)
n
2
3
we have
x 0; when x 2
we have
n 1
(1) 1
n
2n
1
1
n
2n
1
1
3
, a
n 1
( 1)
n 1
2n
1
divergent series
(a) the radius is 1 3 ; the interval of convergence is 2
3
(b) the interval of absolute convergence is
2
3
x 0
(c) the series converges conditionally at x 2
3
, a
conditionally convergent series; when x 0
x
0
33.
un
1
1
x
n 2 4 6 (2 n) 1
n
un
n 2 4 6 (2 n) 2( n 1) n
x
n
2n
2
lim 1 lim 1 x lim 1 for all x
(a) the radius is ; the series converges for all x
(b) the series converges absolutely for all x
(c) there are no values for which the series converges conditionally
34.
n 2
2
2
u 1
3 5 7 (2 1) 2( 1) 1 n
n
n n x
2
(2 3)
lim 1 lim
n
n n
1 x lim 1
2 n 1 n 1 2
n
n
u
n ( n 1) 2 3 5 7 (2n 1) x n 2( n 1)
only x 0 satisfies this inequality
(a) the radius is 0; the series converges only for x 0
(b) the series converges absolutely only for x 0
(c) there are no values for which the series converges conditionally
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