Question

In Problems 22–25, determine whether each infinite geometric series converges or diverges. If it converges, find its sum

$2-1+\frac{1}{2}-\frac{1}{4}+.....$

Short answer

The geometric series converges.

Its sum is:$\frac{4}{3}$

Step-by-step solution

Step 1. Given information.

The series is :

$2-1+\frac{1}{2}-\frac{1}{4}+.....$

Step 2. To find common ratio and first term. 

First term of the series is:$a=2$

Common ratio is:

$r=\frac{-1}{2}$

$\left|r\right|=\frac{1}{2}$

Since$\left|r\right|<1$

Hence given series converges.

Step 3. To find sum of infinite terms. 

$\begin{array}{rcl}{S}_{\infty }& =& \frac{2}{1-\left(-{\displaystyle \frac{1}{2}}\right)}\\ & =& \frac{2}{1+{\displaystyle \frac{1}{2}}}\\ & =& \frac{2}{{\displaystyle \frac{3}{2}}}\\ & =& \frac{4}{3}\end{array}$