   Chapter 11.2, Problem 60E

Chapter
Section
Textbook Problem

Find the values of x for which the series converges. Find the sum of the series for those values of x. ∑ n = 0 ∞ ( − 4 ) n ( x − 5 ) n

To determine

To find:

The values of x for which the series converges, and then, find the sum of the series for those values of x

Explanation

1) Concept:

Use a geometric series to find the values of x for which the series converges then find the sum of the series

Geometric Series:

n=1arn-1=a+ar+ar2+

is convergent if r<1 and its sum is

n=1arn-1=a1 - rr<1

2) Given:

n=0-4nx-5n

3) Calculation:

Consider,

n=0-4nx-5n=n=0-4x+20n

This is a geometric series with initial term a=1 and common ratio r=-4x+20.

By using the concept

The given series converges when r=-4x+20<1

-1< -4x+20<1………

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