Thomas Calculus 13th [Solutions]

752 Chapter 10 Infinite Sequences and Series
10.7 POWER SERIES
1.
u
n 1
n 1
lim 1 lim x 1 | | 1 1 1;
u
n
n
n
n
x
x x when x 1 we have
n
n
( 1) ,
1
a divergent series;
when x 1 we have 1, a divergent series
n 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) there are no values for which the series converges conditionally
2.
n 1
un
1 ( x 5)
lim 1 lim 1 | x 5 | 1 6 x 4; when x 6 we have
u
n
n n n ( x 5)
n
n
( 1) ,
1
a divergent
series; when x 4 we have 1, a divergent series
n 1
(a) the radius is 1; the interval of convergence is 6 x 4
(b) the interval of absolute convergence is 6 x 4
(c) there are no values for which the series converges conditionally
3.
n 1
lim un
1 1 lim (4x
1) 1 | 4 x 1| 1 1 4 x 1 1 1 x
u
(4 1)
2
0; when x 1 we have
n
n n n x
2
n n
( 1) ( 1)
2n ( 1)
n
1 , a divergent series; when x 0 we have
n 1 n 1 n 1
a divergent series
(a) the radius is 1 4 ; the interval of convergence is 1
2
x 0
(b) the interval of absolute convergence is 1
2
x 0
(c) there are no values for which the series converges conditionally
n
n n n
( 1) (1) ( 1) ,
1 n 1
4.
n 1
lim un
1 1 lim (3x 2) n
1 1 3 x 2 lim n
(3 2)
1
1 3 x 2 1 1 3 x
u 2 1
n
n n n
n x
n
n
1
3
x 1; when x 1 we have
3
n
n
( 1)
n
1
which is the alternating harmonic series and is conditionally
convergent; when x 1 we have 1,
the divergent harmonic series
n
n 1
(a) the radius is 1 3 ; the interval of convergence is 1 3
x 1
(b) the interval of absolute convergence is 1 3
x 1
(c) the series converges conditionally at x 1
3
5.
lim 1
u 1
1 lim ( 2) n n
n
x 10 1 x 2
u
1
10
1 2 10 10 2 10 8 12;
n
n
n
n
n
10 ( x 2)
x x x when
n
x 8 we have ( 1) , a divergent series; when x 12 we have
n 1
(a) the radius is 10; the interval of convergence is 8 x 12
(b) the interval of absolute convergence is 8 x 12
(c) there are no values for which the series converges conditionally
n
1,
1
a divergent series
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