   Chapter 11.9, Problem 2E

Chapter
Section
Textbook Problem

Suppose you know that the series ∑ n = 0 ∞ b n x n converges for | x | < 2 . What can you say about the following series? Why? ∑ n = 0 ∞ b n n + 1 x n + 1

To determine

To find:

The radius of convergence of the series

n=0bnn+1xn+1

Explanation

1) Concept:

If the power series cnx-an has radius of convergence R>0, then the function f defined by

fx=c0+c1x-1+c2x-12+=n=0cnx-an is differentiable (and therefore continuous) on the interval (a-R, a+R) and

fxdx=C+c0x-a+c1x-a22+c2x-a33+=C+n=0x-an+1n+1

has the radius of convergence R.

2) Given:

The series n=0bnxn converges for x<2

3) Calculation:

Given that the series n=0bnxn converges for x<2

If fx=bnxn then f(x)=bnxn+1n+1

Therefore, from theorem 2,

If the power series cn

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