Chapter 12, Problem 12CT

To determine

Whether the infinite geometric series $256-64+16-4+\mathrm{...}$ converges or diverges and find its sum if converges.

Answer to Problem 12CT

Solution:

The series $256-64+16-4+\mathrm{...}$ converges and sum of the series is $\frac{1024}{5}$.

Explanation of Solution

Given information:

Infinite geometric series, $256-64+16-4+\mathrm{...}$

Explanation:

Consider the geometric series $256-64+16-4+\mathrm{...}$

Find ratio of consecutive terms and first term.

$r=\frac{-64}{256}=-\frac{1}{4}$ and $a_1=256$.

$\left|r\right|=\left|-\frac{1}{4}\right|=\frac{1}{4}<1$.

Hence, the series converges and the sum is given by:

$S=\frac{a_1}{1-r}=\frac{256}{1-\left(-\frac{1}{4}\right)}=\frac{256}{\frac{5}{4}}=\frac{1024}{5}$

Thus, the series $256-64+16-4+\mathrm{...}$ converges and sum of the series is $\frac{1024}{5}$.