By the root test, the series converges. We have
[tex]\displaystyle\lim_{k\to\infty}\sqrt[k]{\left|(-1)^k\frac{k^3}{3^k}\right|}=\lim_{k\to\infty}\frac{\sqrt[k]{k^3}}3=\frac13\lim_{k\to\infty}k^{\frac3k}=\frac13[/tex]
which is less than 1.
In case it's not clear why [tex]k^{3/k}[/tex] should converge to 1:
[tex]\displaystyle\lim_{k\to\infty}k^{\frac3k}=\lim_{k\to\infty}\exp\left(\ln k^{\frac3k}\right)=\exp\left(\lim_{k\to\infty}\frac{3\ln k}k\right)\to e^0=1[/tex]
