Answer
Converges
Work Step by Step
Apply Root Test such as: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
which states that the series converges when $L \lt 1$; the series diverges when $L \gt 1$
Let us consider $a_n=\dfrac{(n!)^n}{(n^{n^2})^2}$
This implies that $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{(n!)^n}{(n^{n^2})^2}|)^{1/n}$
$L=\lim\limits_{n \to \infty} \dfrac{n!}{ (n)^n}=\lim\limits_{n \to \infty} \dfrac{n (n-1)(n-2) .....1}{(n)^n} =0 \lt 1$
Thus, the series converges by the Root Test.