Short Answer

Find the values of x for which the series $\sum_{k = 0}^{\infty} {(\frac{\cos x}{2})}^{k}$ converges.

The series $\sum_{k = 0}^{\infty} {(\frac{\cos x}{2})}^{k}$ converges for all values of x.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Given a series $\sum_{k = 0}^{\infty} {(\frac{\cos x}{2})}^{k}$ .

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

A geometric series is of the form $\sum_{k = 0}^{\infty} c r^{k}$ for some constants c and r.

Suppose r is a non-zero real number, then $\sum_{k = 0}^{\infty} c r^{k}$ converges to $\frac{c}{1 - r}$ if and only if $|r| < 1$ .

Here, the series role="math" localid="1648833439638" $\sum_{k = 0}^{\infty} {(\frac{\cos x}{2})}^{k}$ has role="math" localid="1648833160702" $r = \frac{\cos x}{2}$ .

For the series to converge, role="math" localid="1648833416407" $|\frac{\cos x}{2}| < 1$ .

Note that $|\cos x| \leq 1$ for all x.

It follows that $|\frac{\cos x}{2}| < 1$ for all values of x.

It follows that $\sum_{k = 0}^{\infty} {(\frac{\cos x}{2})}^{k}$ converges for all values of x.

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Calculus

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Short Answer

Find the values of x for which the series $\sum_{k = 0}^{\infty} {(\frac{\cos x}{2})}^{k}$ converges.

Step by Step Solution

Step 1. Given information.

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

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Find the values of x for which the series $\sum_{k = 0}^{\infty} {(\frac{\cos x}{2})}^{k}$ converges.