Answer
$$ \frac{4}{3}$$
Work Step by Step
Given $$\frac{4}{9}+\frac{8}{27}+\frac{16}{81}+\frac{32}{243}+\cdots$$
This is a geometric series with $|r|= \frac{8}{27}\frac{9}{4}=\frac{2}{3}<1$, so the series converges and has the sum
\begin{align*}
S&=\frac{a}{1-r}\\
&=\frac{4/9}{1-\frac{2}{3}}\\
&= \frac{4}{3}
\end{align*}
