James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 11.8 Power Series 747
In fact if we put x − 1 2 in the geometric series (2) we get the convergent series
n−0S ò 2D
1 n
− 1 1 1 2 1 1 4 1 1 8 1 1 16 1 ∙ ∙ ∙
but if we put x − 2 in (2) we get the divergent series
More generally, a series of the form
ò 2 n − 1 1 2 1 4 1 8 1 16 1 ∙ ∙ ∙
n−0
3
ò c n sx 2 ad n − c 0 1 c 1 sx 2 ad 1 c 2 sx 2 ad 2 1 ∙ ∙ ∙
n−0
Notice that
sn 1 1d! − sn 1 1dnsn 2 1d ? . . . ? 3 ? 2 ? 1
− sn 1 1dn!
is called a power series in sx 2 ad or a power series centered at a or a power series
about a. Notice that in writing out the term corresponding to n − 0 in Equations 1 and 3
we have adopted the convention that sx 2 ad 0 − 1 even when x − a. Notice also that
when x − a, all of the terms are 0 for n > 1 and so the power series (3) always converges
when x − a.
Example 1 For what values of x is the series ò n!x n convergent?
n−0
SOLUtion We use the Ratio Test. If we let a n , as usual, denote the nth term of the
series, then a n − n!x n . If x ± 0, we have
lim
nl`
Z
a n11
a n
Z − lim
nl`
Z
sn 1 1d!x n11
n!x n
Z − lim
nl`
sn 1 1d| x | − `
By the Ratio Test, the series diverges when x ± 0. Thus the given series converges only
when x − 0.
n
sx 2 3d
Example 2 For what values of x does the series ò
n
n−1 n
SOLUtion Let a n − sx 2 3d n yn. Then
Z
a n11
a n
Z − Z
sx 2 3d n11
n 1 1
?
n
sx 2 3d n
Z
converge?
− 1
1 1 1 n
| x 2 3 | l | x 2 3 | as n l `
By the Ratio Test, the given series is absolutely convergent, and therefore convergent,
when | x 2 3 | , 1 and divergent when | x 2 3 | . 1. Now
| x 2 3 |
, 1 &? 21 , x 2 3 , 1 &? 2 , x , 4
so the series converges when 2 , x , 4 and diverges when x , 2 or x . 4.
The Ratio Test gives no information when | x 2 3 | − 1 so we must consider x − 2
and x − 4 separately. If we put x − 4 in the series, it becomes o 1yn, the harmonic
series, which is divergent. If x − 2, the series is o s21d n yn, which converges by the
Alternating Series Test. Thus the given power series converges for 2 < x , 4. n
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