Answer
Converges
Work Step by Step
Apply Root Test such as: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
This test states that when $L \lt 1$; the series converges and for $L \gt 1$; the series diverges.
Let us consider $a_n=\dfrac{n}{(\ln n)^{n/2}}$
$L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{n}{(\ln n)^{(n/2)}}|)^{(1/n)}$
$\implies L=\lim\limits_{n \to \infty} \dfrac{\sqrt [n] n}{ \sqrt {\ln (n)}}=\dfrac{1}{\infty}=0 \lt 1$
Hence, the series converges by the Root Test.