Answer
Convergent
Work Step by Step
$\lim\limits_{n \to \infty}\sqrt[n] a_{n}=\lim\limits_{n \to \infty}\sqrt[n] \frac{n^{2}}{e^{n^{3}}}$
$=\frac{1}{\lim\limits_{n \to \infty}(e^{n^{3}})^{1/n}}$
$=\frac{1}{\lim\limits_{n \to \infty}(e^{n^{2}})}$
$=0 \lt 1$
The series is convergent by root test.