Answer
$\dfrac{1024}{5}$
Work Step by Step
An infinite geometric series converges if and only if $|r|\lt1$, where $r$ is the common ratio.
If it converges, then its sum equals $\frac{a_1}{1-r}$ where $a_1$ is the first term.
The common ratio is the quotient of two consecutive terms.
Thus, the given series has:
$r=\dfrac{a_2}{a_1}=\dfrac{-64}{256}=-\dfrac{1}{4}$
$\left|-\dfrac{1}{4}\right|=\dfrac{1}{4}\lt1$, thus the series converges.
Hence, with $a_1=256$, the sum is:
$\dfrac{256}{1-(-\frac{1}{4})}=\dfrac{256}{1+\frac{1}{4}}=\dfrac{256}{\frac{5}{4}}=256 \cdot \dfrac{4}{5}=\dfrac{1024}{5}$