Answer
Converges, sum: $\frac{8}{5}$
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 it equals $\frac{a_1}{1-r}$ where $a_1$ is the first term.
The common ratio is the quotient of two consecutive terms: $r=\frac{a_2}{a_1}=\dfrac{-\frac{1}{2}}{2}=-\frac{1}{4}$. $|-\frac{1}{4}|=\frac{1}{4}\lt1$, thus it converges.
Hence the sum (since $a_1=2$): $\dfrac{2}{1-(-\frac{1}{4})}=\frac{8}{5}$