Answer
$S_{\infty}=\dfrac{16}{9}$
Work Step by Step
Using $S_{\infty}=\dfrac{a_1}{1-r}$, the sum of the infinite geometric sequence $
2,-\dfrac{1}{4},\dfrac{1}{32},...
$ is
\begin{array}{l}
S_{\infty}=\dfrac{2}{1-\left( -\dfrac{1}{8} \right)}\\\\
S_{\infty}=\dfrac{2}{\dfrac{9}{8}}\\\\
S_{\infty}=\dfrac{16}{9}
.\end{array}