   Chapter 11.8, Problem 3E

Chapter
Section
Textbook Problem

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

To determine

To find:

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

n=1-1nnxn

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 ratio test state that if limnan+1an<1 then the series n=1an converges.

2) Given:

n=1-1nnxn

3) Calculation:

The given series is

n=1-1nnxn

So, the nth term is

an=-1nnxn

Therefore,

limnan+1an=limn-1n+1n+1xn+1-1nnxn

=limn-n+1xn

=

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