   Chapter 11.8, Problem 40E

Chapter
Section
Textbook Problem

# Suppose that the power series ∑ cn(x − a)n satisfies cn ≠ 0 for all n, Show that if limn→∞|cn/cn+1| exists, then it is equal to the radius of convergence of the power series.

To determine

To show: If limn|cncn+1| exists, then it is equal to the radius of convergence of the power series n=0cn(xa)n with cn0 .

Explanation

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

Calculation:

Let an=cn(xa)n .

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

Consider a power series n=0cn(xa)n with cn0 for all n and limn|cncn+1|=cR .

Case 1: c0

Obtain |an+1an| .

|an+1an|=|cn+1(xa)n+1cn(xa)n|

Take limn on both sides,

limn|an+1an|=limn(|cn+1(xa)n+1cn(xa)n|)=limn(|cn+1cn||xa|)

limn|an+1an|=|xa|limn|cn+1cn| (1)

Since limn|cncn+1|=c , limn|cn+1cn|=1c

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