Chapter 9.1, Problem 94E

To determine

To find the sum of the infinite series ${\displaystyle \sum _{k=1}^{\infty }\frac{1}{{10}^k}}$

Answer to Problem 94E

  ${\displaystyle \sum _{k=1}^{\infty }\frac{1}{{10}^k}}=\frac{1}{9}$

Explanation of Solution

Given information :

Given series ${\displaystyle \sum _{k=1}^{\infty }\frac{1}{{10}^k}}$

As we have ${\displaystyle \sum _{k=1}^{\infty }\frac{1}{{10}^k}}$

  $\begin{array}{l}=\left(\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10000}+\mathrm{..........}\right)\\ =\left(0.1+0.01+0.001+0.0001+\mathrm{..........}\right)\\ =\left(\mathrm{0.11111.........}\right)\\ =\frac{1}{9}\end{array}$

Therefore, ${\displaystyle \sum _{k=1}^{\infty }\frac{1}{{10}^k}}=\frac{1}{9}$