Chapter 9, Problem 60RE

To determine

Find the sum of the given infinite geometric series

Answer to Problem 60RE

The sum of the infinite geometric series is 2.

Explanation of Solution

Given:

The geometric sequence,

  ${\displaystyle \sum _{i=1}^{\infty }{\left(0.5\right)}^{i-1}}$

Consider the given geometric sequence,

  ${\displaystyle \sum _{i=1}^{\infty }{\left(0.5\right)}^{i-1}}$

Write down first few terms

  $1+0.5+{\left(0.5\right)}^2+\mathrm{....}$

First find out common ratio $r$

The common ratio $r$ is can be found out by dividing any term after the first term that directly precedes it.

In above geometric sequence,

  $a_1=1,{\text{ a}}_2=0.5$

Then,

  $\begin{array}{l}r=\frac{a_2}{a_1}\hfill \\ =\frac{0.5}{1}\hfill \\ =0.5\hfill \end{array}$

Since, $r=0.5$ the condition that $\left|r\right|<1$ is met.

Thus, the infinite geometric series have sum.

The sum $S$ infinite geometric series having common ratio $r$ and first term $a_1$ is,

  $S=\frac{a_1}{1-r}for\left|r\right|<1$

Then,

  $\begin{array}{l}S=\frac{1}{1-0.5}\hfill \\ =\frac{1}{0.5}\hfill \\ =2\hfill \end{array}$

Hence, the sum of the infinite geometric series is 2.