   Chapter 11.2, Problem 61E

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.61. ∑ n = 0 ∞ 2 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=02nxn.

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=02nxn=n=0(2x)n=(2x)0+(2x)1+(2x)2+(2x)3+=1+(2x)+(2x)2+(2x)3+

Clearly, it is geometric series with first term of the series is a=1 and common ratio is r=2x.

Use the Result stated above, the geometric series n=02nxn is converges if |r|<1

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