   Chapter 11.2, Problem 57E

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.57. ∑ n = 1 ∞ ( − 5 ) n x n

To determine

To find: The values of x if the given series is convergent and obtain the sum of the series.

Explanation

Given:

The series is n=1(5)nxn.

Result used:

The geometric series n=1arn1 (or) a+ar+ar2+ is converges if |r|<1 and its sum is a1r, where a is the first term and r is the common ratio of the series.

Calculation:

Obtain the value of x (the interval of converges).

The given series can be expressed as follows,

n=1(5)nxn=n=1(5x)n=(5x)+(5x)2+(5x)3+(5x)4+

Clearly, it is geometric series with first term of the series is a=5x and common ratio is r=5x

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