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Chapter 7: Q 70. (page 615)

Find the values of x for which the series 鈭�k=0鈭�cosx2kconverges.

Short Answer

Expert verified

The series 鈭�k=0鈭�cosx2kconverges for all values ofx.

Step by step solution

01

Step 1. Given information.

Given a series 鈭�k=0鈭�cosx2k.

02

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

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

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

Here, the series role="math" localid="1648833439638" 鈭�k=0鈭�cosx2khas role="math" localid="1648833160702" r=cosx2.

For the series to converge, role="math" localid="1648833416407" cosx2<1.

Note that cosx鈮�1 for allx.

It follows that cosx2<1 for all values of x.

It follows that 鈭�k=0鈭�cosx2kconverges for all values of x.

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