Thomas Calculus 13th [Solutions]

790 Chapter 10 Infinite Sequences and Series
44.
lim 1
1
1 lim n
u
2 (2 1)
n
n
n x 2n 1 2
2 1 2 2 1 2 1
2 3 1 1 1 x
2 lim n n
2 3 1 1 x
u 2
(1) 1
n
n
n
n n n n
n n 2 (2x
1)
n
3 1
2 2
2x 1 2 2 2x 1 2 3 2x 1 x ; at x 3
we have
n
n 1 ( 2) ( 1) ( n 1)
2n
1 n
2
2n
1
n 1 n 1
n
which diverges by the nth-Term Test for Divergence since lim 1 1
0;
2
n
n
2n
1 2
x we have
n 1 2 n 1
, which diverges by the nth-Term Test
at
1
2
n
2n
1 n
2 2n
1
n 1 n 1
(a) the radius is 1; the interval of convergence is
3 x 1
2 2
(b) the interval of absolute convergence is
3 x 1
2 2
(c) there are no values for which the series converges conditionally
45.
u n
x n n
x x
n n ( n 1) x
n n
lim
1
1
1 lim n n
n
1 1
u 1 lim 1
1 1 1 lim 1
1 0 1,
n n
n
n n e n e
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
x which holds
46.
u
lim 1
1
1 lim n
n
x n
1 lim n
u
1
1 1;
n
n
n
n n n 1 x
n
x x when x 1 we have
n
( 1)
n
1
n
,
which
converges by the Alternating Series Test; when x 1 we have
1
,
(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
n 1
n
a divergent p-series
47.
lim 2 1
u 2
1
1 lim ( 2) n
n
n
n x 3 1 x 2
1 2 1 3 lim n
1
1 3 x
u
3;
n
n
n
n
n n 3 ( n 1) x
n
the series
n 1
and
n 1,
n 1
3
n 1
3
obtained with x 3, both diverge
(a) the radius is 3; the interval of convergence is 3 x 3
(b) the interval of absolute convergence is 3 x 3
(c) there are no values for which the series converges conditionally
48.
lim 2 3
un
1
1 lim ( x 1) x n
2n 1 2 2 1
2 2
2 3 1 ( x 1) lim n
2 1
2 3
1 ( x 1) (1) 1 ( x
u 1) 1
n
n
n n
n n ( x 1) x
n
| x 1| 1 1 x 1 1 0 x 2; at x 0 we have
n 2n 1 3n 1 n 1
( 1) ( 1) ( 1) ( 1)
2n 1 2n 1 2n
1
n 1 n 1 n 1
which
converges conditionally by the Alternating Series Test and the fact that
1
n 2n 1
n
( 1) (1) ( 1)
2n
1 2n
1
n 1 n 1
,
which also converges conditionally
(a) the radius is 1; the interval of convergence is 0 x 2
(b) the interval of absolute convergence is 0 x 2
(c) the series converges conditionally at x 0 and x 2
2 1
n 1 n
diverges; at x 2 we have
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