Question

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

$3+1+\frac{1}{3}+\frac{1}{9}+......$

Short answer

The geometric series converges.

Its sum is:$\frac{9}{2}$

Step-by-step solution

Step 1. Given information

The geometric series:

$3+1+\frac{1}{3}+\frac{1}{9}+.....$

Step 2. To find common ratio and first term.

First term of the series is:$a_1=3$

Common ratio is:$\frac{1}{3}$

Since common ratio of the series is less than 1.

Hence the series converges.

Step 3. To find sum of infinite terms.

$\begin{array}{rcl}{S}_{\infty }& =& \frac{3}{1-{\displaystyle \frac{1}{3}}}\\ & =& \frac{3}{{\displaystyle \frac{2}{3}}}\\ & =& \frac{3\times 3}{2}\\ & =& \frac{9}{2}\end{array}$