Answer
Converges
Work Step by Step
Let us consider $a_n=(-1)^n\dfrac{n^2(n+2)}{( n!) 3^{(2n)}}$
In order to solve the given series we will take the help of Ratio Test. This test states that when the limit $L \lt 1$, the series converges and for $L \gt 1$, the series diverges.
$L=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)^2(n+3)}{ (n+1)! 3^{2n+2}}}{\dfrac{n^2(n+2)}{ n! 3^{2n}}}|$
$\implies \lim\limits_{n \to \infty}|\dfrac{(n+1)^2 (n+3)}{(9) n^2 (n+1)}|=\dfrac{1}{9}$
so, $L \lt 1$
Thus, the series Converges by the ratio test.