Chapter 7.4, Problem 17E

Solution

To calculate : The sum of infinite geometric series ${\displaystyle \sum _{i=0}^{\infty }{\left(-\frac{3}{7}\right)}^i}$ .

The sum of geometric series ${\displaystyle \sum _{i=0}^{\infty }{\left(-\frac{3}{7}\right)}^i}$ is $\frac{7}{10}$ .

Given information : The geometric series ${\displaystyle \sum _{i=0}^{\infty }{\left(-\frac{3}{7}\right)}^i}$ .

Formula used : The sum of infinite geometric series with first term $a_1$ and common ration $r$ is calculated using the formula $S=\frac{a_1}{1-r}$ .

Here, $\left|r\right|<1$ . If $\left|r\right|\ge 1$ , then the series has no sum.

Calculation :

Consider the geometric series ${\displaystyle \sum _{i=0}^{\infty }{\left(-\frac{3}{7}\right)}^i}$ ,

The series can be written as ${\displaystyle \sum _{i=1}^{\infty }1\cdot {\left(-\frac{3}{7}\right)}^{i-1}}$ .

For the given series the value of $a_1$ is $1$ and $r$ is $-\frac{3}{7}$ .

As the common ratio $\left|-\frac{3}{7}\right|=\frac{3}{7}<1$ , the sum of the series exists.

Then the sum of infinite geometric series is calculated using the formula $S=\frac{a_1}{1-r}$ .

  $\begin{array}{c}S=\frac{1}{1-\left(-\frac{3}{7}\right)}\\ =\frac{1}{1+\frac{3}{7}}\\ =\frac{1}{\frac{10}{7}}\\ =\frac{7}{10}\end{array}$

Thus, the sum of geometric series ${\displaystyle \sum _{i=0}^{\infty }{\left(-\frac{3}{7}\right)}^i}$ is $\frac{7}{10}$ .