Chapter 8.4, Problem 11E

To determine

Find the sum of the given infinite geometric series.

Answer to Problem 11E

Thesum of the series is $8$ .

Explanation of Solution

Given:

The given infinite geometric seriesis $2+\frac{6}{4}+\frac{18}{16}+\frac{54}{64}+\cdot \cdot \cdot$ .

Calculation:

The sum of an infinite geometric series with first term $a_1$ and common ratio $r$ is

  $S=\frac{a_1}{1-r}$ , where $r<1$ .

If $r\ge 1$ , then the series has no sum.

In the given series $a_1=2$ and

  $\begin{array}{l}r=\frac{\frac{6}{4}}{2}\\ =\frac{6}{4}\cdot \frac{1}{2}\\ =\frac{3}{4}<1\end{array}$ .

  $\begin{array}{l}S=\frac{2}{1-\frac{3}{4}}\\ =\frac{2}{\frac{1}{4}}\\ =2\cdot 4\\ =8\end{array}$

Hence the sum of the series is $8$ .