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