Answer
Divergent
Work Step by Step
We are given the series:
$3-4+\dfrac{16}{3}-\dfrac{64}{9}+....$
We determine the ratios of consecutive terms:
$\dfrac{-4}{3}=-\dfrac{4}{3}$
$\dfrac{\dfrac{16}{3}}{-4}=-\dfrac{16}{3}\cdot\dfrac{1}{4}=-\dfrac{4}{3}$
$\dfrac{-\dfrac{64}{9}}{\dfrac{16}{3}}=-\dfrac{64}{9}\cdot\dfrac{3}{16}=-\dfrac{4}{3}$
As all ratios are the same, the series is a geometric series. Its ratio is
$r=-\dfrac{4}{3}$.
Because $|r|=\left|-\dfrac{4}{3}\right|=\dfrac{4}{3}>1$, the series is divergent.