Answer
$S_{\infty}=\dfrac{4}{9}$
Work Step by Step
Using $S_{\infty}=\dfrac{a_1}{1-r}$, the sum of the infinite geometric sequence $
\dfrac{2}{3},-\dfrac{1}{3},\dfrac{1}{6},...
$ is
\begin{array}{l}
S_{\infty}=\dfrac{\dfrac{2}{3}}{1-\left( -\dfrac{1}{2} \right)}\\\\
S_{\infty}=\dfrac{\dfrac{2}{3}}{\dfrac{3}{2}}\\\\
S_{\infty}=\dfrac{4}{9}
.\end{array}