Question

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

Short answer

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

Step-by-step solution

Step 1. Given information.

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

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.

Supposer 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" $\sum _{k=0}^{\infty }{\left(\frac{\cos x}{2}\right)}^k$has role="math" localid="1648833160702" $r=\frac{\cos x}{2}$.

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

Note that $\left|\cos x\right|\le 1$ for allx.

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

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