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Q 70.

Expert-verifiedFound in: Page 615

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*.

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

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*.

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