Chapter 11.6, Problem 8WE

To determine

To calculate:

Sum of given geometric series.

Answer to Problem 8WE

Sum of given geometric series is $0.3$

Explanation of Solution

Give information:

  $\frac{1}{2}-\frac{1}{3}+\frac{2}{9}-\frac{4}{27}+\mathrm{...}$

Calculation:

In order to find out common ratio of given number,

  $\frac{1}{2}-\frac{1}{3}+\frac{2}{9}-\frac{4}{27}+\mathrm{...}$

Here first term is $a_1=\frac{1}{2}$

So common ratio is $r=\frac{-1/3}{1/2}=-\frac{2}{3}$

Since $\left|r\right|=\left|-\frac{2}{3}\right|<1$

Infinite sum exists.

So, Sum of given geometric series is calculated as,

  $=\frac{a^n}{1-r}$

Where,

  $a_1=$ Initial term.

  $r=$ common ratio between terms.

  $n=$ number of terms

Put the values in the equation.

  $\begin{array}{l}=\frac{a^n}{1-r}\\ =\frac{1/2}{1-\left(-\frac{2}{3}\right)}\\ =\frac{1/2}{1+\left(\frac{2}{3}\right)}\\ =\frac{1/2}{\frac{5}{3}}\\ =\frac{\left(3\right)\left(1/2\right)}{5}\\ =\frac{1.5}{5}\\ =0.3\end{array}$

Therefore, sum of given geometric series is $0.3$ .