Chapter 9, Problem 18RE

To determine

To find : The sum of the given infinite series.

Explanation of Solution

Given: The series is ${\displaystyle \sum _{k=1}^{\infty }2{\left(\frac{1}{100}\right)}^k}$

Consider the given series.

  ${\displaystyle \sum _{k=1}^{\infty }2{\left(\frac{1}{100}\right)}^k}$

The first term of the series is,

  $a=\frac{2}{100}$

The common difference is,’

  $r=\frac{1}{100}$

The sum of the series is calculated as,

  $\begin{array}{c}\text{Sum}=\frac{a}{1-r}\\ =\frac{\frac{2}{100}}{1-\frac{1}{100}}\\ =\frac{2}{99}\end{array}$

Therefore, the sum of the series is $\frac{2}{99}$ .