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Chapter 7: Q 67. (page 615)

Find the values of x for which the series鈭�k=0鈭�xkconverges.

Short Answer

Expert verified

The series 鈭�k=0鈭�xkconverges only for-1<x<1.

Step by step solution

01

Step 1. Given information.

Given a series鈭�k=0鈭�xk.

02

Step 2. Find all values of x for which the series converges.

A geometric series is of the form 鈭�k=0鈭�crk for some constants c and r.

Supposer is a non-zero real number, then 鈭�k=0鈭�crkconverges to c1-rif and only if r<1.

Here, the serieslocalid="1648830940128" 鈭�k=0鈭�xkhas r=x.

For the series to converge, x<1.

Note that ify<a, it follows that-a<y<a.

It follows that 鈭�k=0鈭�xkconverges for all -1<x<1.

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