Answer
The series converges by the Ratio Test.
Work Step by Step
Use the Ratio Test:$\lim\limits_{n \to \infty}\Big|\frac{a_{n+1}}{a_n}\Big|$.
Let $a_n=\frac{3^nn^2}{n!}$. Then, $\lim\limits_{n\to \infty}\Bigg|\frac{3^{n+1}(n+1)^2}{(n+1)!} \cdot \frac{n!}{3^nn^2}\Bigg|$. Because $n!$, $n^2$, and $3^n$ are positive for all $n>0$, we can remove the absolute value lines. Now, $\lim\limits_{n \to \infty}\Bigg[\frac{3^{n+1}(n+1)^2}{(n+1)!} \cdot \frac{n!}{3^nn^2}\Bigg]= \lim\limits_{n \to \infty}\Bigg[\frac{3(n+1)^2}{(n+1)n^2}\Bigg]=$
$3\cdot\lim\limits_{n \to \infty}\Bigg[\frac{n+1}{n^2}\Bigg]\to 0$ as $n\to \infty$
Therefore, the limit is $(3)(0)=0<1$, and the series converges by the Ratio Test.
