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 $a_n=\dfrac{(n^n)}{(2)^{n^2}}$
Now, we can see that $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{(n^n)}{(2)^{n^2}}|)^{1/n}$
$\implies \lim\limits_{n \to \infty} \dfrac{n}{(2)^n}=0 \lt 1$
Hence, the series converges by the Root Test.