   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 describe: The series n=0bnn+1.xn+1 converges.

Explanation

Given:

The series n=0bnn+1.xn+1 is converges or not .if the series n=0bnxn is converges |x|<2

Theorem used:

If the power series n=0cn(xa)n has radius of convergence R>0 and is integrable on the interval (aR,a+R) , then the function f(x)dx=C+n=1cn(xa)n+1n+1 is converges to R.

Calculation:

Let f(x)=n=0bnxn .

f(x)=b0x0+b1x1+b2x2++bnxn+=1+b1x+b2x2++bnxn+

The integration of f(x) is computed as follows,

f(x)dx=(1+b1x+b2x2++bnx<

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