Answer
Converges
Work Step by Step
Consider $a_n=\dfrac{(n!)^n}{(n^{n^2})}$
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{(n!)^n}{(n^{n^2})}|)^{1/n}=\lim\limits_{n \to \infty} \dfrac{n!}{ n^n}$
So, $l=\lim\limits_{n \to \infty} \dfrac{n (n-1)(n-2) .....1}{n^n} =0 \lt 1$
Thus, the given series Converges by the Root Test.