   Chapter 11.2, Problem 58E

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 = 1 ∞ ( x + 2 ) 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=1x+2n

3) Calculation:

Consider

n=1x+2n

This is a geometric series with initial term a=x+2 and common ratio r=x+2

By using a geometric series

The given series converges when r=x+2<1

-1<x+2<1………

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