Q 70. - Calculus

Question

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

Step-by-Step Solution

Answer

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

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.

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

For the series to converge, $\left|\frac{\cos x}{2}\right|<1$.

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

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.