Answer
$S_{\infty}=\dfrac{1}{9}$
Work Step by Step
Using $S_{\infty}=\dfrac{a_1}{1-r}$, the sum of the infinite geometric sequence $
\dfrac{1}{10},\dfrac{1}{100},\dfrac{1}{1000}...
$ is
\begin{array}{l}
S_{\infty}=\dfrac{\dfrac{1}{10}}{1-\dfrac{1}{10}}\\\\
S_{\infty}=\dfrac{\dfrac{1}{10}}{\dfrac{9}{10}}\\\\
S_{\infty}=\dfrac{1}{9}
.\end{array}