Answer
Diverges
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 $a_n=\dfrac{(n!)^n}{(n^n)^2}$
This implies that $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{(n!)^n}{(n^n)^2}|)^{(1/n)}=$
and $L=\lim\limits_{n \to \infty} \dfrac{n!}{(n^2)}=\infty \gt 1$
Hence, the series diverges by the Root Test.