   Chapter 11.9, Problem 1E

Chapter
Section
Textbook Problem

If the radius of convergence of the power series ∑ n = 0 ∞ c n x n is 10, what is the radius of convergence of the series ∑ n = 1 ∞ n c n x n − 1 ? Why?

To determine

To find:

The radius of convergence of the series

n=1ncnxn-1

Explanation

1) Concept:

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

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

f'x=c1+2c2x-a+3c3x-a2=n=1ncnx-an-1

then the radius of convergence of the power series f'x=n=1ncnx-an-1 has the radius of convergence R.

2) Given:

The radius of convergence of the power series n=0cnxn is 10.

3) Calculation:

Given is the radius of convergence of the power series n=0cnxn which is 10

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