Answer
Converges
Work Step by Step
Given
$$\sum_{n=1}^{\infty} \frac{n^3}{3^{n^2} }$$
By using the Ratio Test, we get:
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty} \frac{(n+1)^3}{3^{n^2+2n+1}} \frac{3^{n^2}}{n^3} \\
&=\lim _{n \rightarrow \infty} \frac{(n+1)^3}{n^{3} }\lim _{n \rightarrow \infty} \frac{1}{3^{2n+1}}\\
&= 0<1
\end{align*}
Thus the series converges.
