   Chapter 11.8, Problem 11E

Chapter
Section
Textbook Problem

Find the radius of convergence and interval of convergence of the series. ∑ n = 1 ∞ ( − 1 ) n 4 n n x n

To determine

To find:

The radius of convergence and the interval of convergence of the series

n=1-1n4nnxn

Explanation

1) Concept:

i) For a power series n=0cnx-an, there is a positive number R such that the series converges if x-a<R and diverges if x-a>R, this number R is called as a radius of convergence. From this, there are four possible cases of interval of convergence

a-R, a+R,  a-R, a+R,   a-R, a+R,  a-R, a+R

ii) The p-series test state that the series n=11/np converges if p>1

iii) The alternating series:

n=1-1n-1bn Converges if bn+1bn and limnbn=0

iv) The ratio test states that if limnan+1an<1, then the series n=1an converges.

2) Given:

n=1-1n4nnxn

3) Calculation:

The given series is n=1-1n4nnxn

Therefore, the nth term is an=-1n4nnxn

Therefore,

limnan+1an=limn-1n+14n+1xn+1n+1·nxn-1n4n

limnan+1an=limn-4x1+1/n

=4x

Therefore, by using the ratio test, this series is convergent if 4x<1 i

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