   Chapter 11.8, Problem 18E

Chapter
Section
Textbook Problem

Find the radius of convergence and interval of convergence of the series. ∑ n = 1 ∞ n 8 n ( x + 6 ) n

To determine

To find:

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

n=1n(x+6)n8n

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 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=1n(x+6)n8n

3) Calculation:

The given series is n=2n(x+6)n8n

Therefore, the nth term is an=n(x+6)n8n

Therefore,

limnan+1an=limnn+1(x+6)n+18n+1·8nn(x+6)n

limnan+1an

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