∞ ∞
For
what values of x does a Power Series Converge? ∑ an (x – b)n =
∑ un
n=0 n=0
|
The radius of convergence
of a power series can be zero, finite, or infinite. Below are three examples
illustrating each case. |
|
Example 1 ∞ ∑ n!
( x
˗ 1 ) n
Here un =
n! ( x ˗ 1 ) n , un + 1 = (n + 1)! ( x ˗ 1 ) n+1 n = 0 lim | (un+1) / un | = lim | (n + 1) (x ˗ 1 ) | <
1 for convergence n → ∞ n →∞ This inequality can only be satisfied
for x = 1. Radius of convergence = 0. |
|
Example 2 (repeated
here for illustration) ∞ ∑
( x ˗ 1 ) n
Here un =
( x ˗ 1 ) n , un + 1 =
( x ˗ 1 ) n+1 n = 0 lim | (un+1) / un | = lim | ( x ˗
1 ) | < 1 for convergence n → ∞ n → ∞ Note both ∑(
˗ 1 ) n and ∑ ( 1 ) n are divergent infinite series. (x=0 and x =2) So 0
< x < 2 Radius of convergence = 1. |
|
∞ Example 3 ∑
( -1) n ( x ˗ 1
) n+1 / [ 1 ∙ 3 ∙ 5 ∙
∙ ∙ ∙ ( 2n – 1)
] n = 1 Here un
= ( x ˗ 1 ) n+1 / [ 1
∙ 3 ∙ 5 ∙ ∙ ∙ ∙ (
2n – 1) ] un + 1 =
( x ˗ 1 ) n+2 / [
1 ∙ 3 ∙ 5 ∙ ∙ ∙ ∙ ( 2n – 1) ( 2n + 1) ] lim | (un+1) / un | = lim | ( x ˗
1 ) / (2n + 1) | <
1 for convergence n → ∞ n → ∞ which gives | x ˗ 1 | lim [ 1 / (2n + 1) ]|
< 1 which holds for all
values of x n
→ ∞ So
˗ ∞ < x
< ∞ Radius of convergence = ∞ . |
Copyright © 2017 Richard C. Coddington
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