Chapter SH, Problem 10.11EP

To determine

To find: The sum of the given infinite series if it exists.

Answer to Problem 10.11EP

The sum of the given infinite series is $\frac{1}{2}$ .

Explanation of Solution

Given:

The given sequence is $\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\mathrm{...}$ .

Calculation:

The formula for the sum of an infinite geometric series with $-1<r<1$ is given by $S=\frac{a_1}{1-r}$ .

Sun will exist if the common ratio is between $-1<r<1$ .

Find the common ratio as follows.

  $\begin{array}{c}\frac{1/9}{1/3}=\frac{1}{3}\\ \frac{1/27}{1/9}=\frac{1}{3}\\ ⋮\end{array}$

The ratio is $\frac{1}{3}$ , whose absolute value is less than 1. So, some will exist.

Substitute $\frac{1}{3}$ for $a_1$ and $\frac{1}{3}$ for $r$ in the equation $S=\frac{a_1}{1-r}$ to find the sum.

  $\begin{array}{c}S=\frac{\frac{1}{3}}{1-\frac{1}{3}}\\ =\frac{\frac{1}{3}}{\frac{2}{3}}\\ =\frac{1}{2}\end{array}$

Therefore, the sum of the given infinite series is $\frac{1}{2}$ .