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Q 70.

Expert-verified
Found in: Page 615

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

# Find the values of x for which the series $\sum _{k=0}^{\infty }{\left(\frac{\mathrm{cos}x}{2}\right)}^{k}$ converges.

The series $\sum _{k=0}^{\infty }{\left(\frac{\mathrm{cos}x}{2}\right)}^{k}$ converges for all values of x.

See the step by step solution

## Step 1. Given information.

Given a series $\sum _{k=0}^{\infty }{\left(\frac{\mathrm{cos}x}{2}\right)}^{k}$.

## 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 $\left|r\right|<1$.

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

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

Note that $\left|\mathrm{cos}x\right|\le 1$ for all x.

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

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