Something tells me the first term is supposed to be [tex]\dfrac13[/tex], not [tex]\dfrac12[/tex]. In that case, the series would be
[tex]\displaystyle\sum_{n=0}^\infty\frac{(-2)^n}{3^{n+1}}=\frac13\sum_{n=0}^\infty\left(-\frac23\right)^n[/tex]
which is geometric with a ratio less than 1, so the infinite series will converge, and moreover will converge to
[tex]\dfrac13\cdot\dfrac1{1+\frac23}=\dfrac15[/tex]
Even if the first term was [tex]\dfrac12[/tex], the series will still converge, but we can't write it as we did above. Not counting the first term, we still have a convergent geometric series.
