Answer
Diverges
Work Step by Step
Consider $a_n=\dfrac{(n^n)}{(2^n)^2}$
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{(n^n)}{(2^n)^2}|)^{1/n}=\lim\limits_{n \to \infty} \dfrac{n}{4}=\lim\limits_{n \to \infty} \dfrac{1}{4/n}$
So, $l=\infty \gt 1$
Thus, the given series Diverges by the Root Test.