Answer
The sum converges and equals to $\frac{8}{3}\approx 2.6667$
Work Step by Step
In a geometric series the ratio between consecutive terms is constant. In order to find this ratio, we simply divide two consecutive terms: $r=\frac{a_2}{a_1}=\frac{\frac{2}{3}}{\frac{4}{3}}=\frac{1}{2}$
The sum of an infinite geometric series exists only if $\mid r\mid<1$, here it is true. Therefore the sum converges.
The sum of an infinite geometric series is:
$S_\infty=\frac{a_1}{1-r}$
Here: $S_\infty=\frac{\frac{4}{3}}{1-\frac{1}{2}}=\frac{\frac{4}{3}}{\frac{1}{2}}=\frac{8}{3}\approx 2.6667$
