Answer
$S_{\infty}=\dfrac{2}{9}$
Work Step by Step
Using $S_{\infty}=\dfrac{a_1}{1-r}$, the sum of the infinite geometric sequence $
0.2,0.02,0.002,...
$ is
\begin{array}{l}
S_{\infty}=\dfrac{0.2}{1-0.1}\\\\
S_{\infty}=\dfrac{0.2}{0.9}\\\\
S_{\infty}=\dfrac{2}{9}
.\end{array}