   Chapter 11.8, Problem 21E

Chapter
Section
Textbook Problem

# Find the radius of convergence and interval of convergence of the series.21. ∑ n = 1 ∞ n b n ( x − a ) n ,   b > 0

To determine
The radius of convergence and interval of convergence of the series.
Explanation

Given:

The series n=1nbn(xa)n,b>0

Result used:

(1) Ratio test: If limn|an+1an|=L<1 , then the series n=1an is absolutely convergent.

(2) Test for divergence: If limnan does not exist or if limnan0 , then the series n=0an is divergent.

Calculation:

Let an=nbn(xa)n

Then, an+1=n+1bn+1(xa)n+1 .

Obtain |an+1an| .

|an+1an|=|n+1bn+1(xa)n+1bnn(xa)n|

Take limn on both sides,

limn|an+1an|=limn|n+1(xa)n+1bn+1bnn(xa)n|=limn|(xa)b|(n+1n)=limn|(xa)b|(1+1n)

Apply the limit and simplify the terms as shown below.

limn|(xa)b|(1+1n)=|(xa)b|(1+1)=|(xa)b|(1+0)=|(xa)b|

The series n=1(xa)n,b>0 converges as |x|<1

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