Answer
Converges
Work Step by Step
Let $u_n=\dfrac{(\ln n)^2}{(n)^{3/2}}$ and $v_n=\dfrac{1}{ (n)^{5/4}}$
Now, $\lim\limits_{n \to \infty}\dfrac{u_n}{v_n} =\lim\limits_{n \to \infty}\dfrac{\dfrac{(\ln n)^2}{(n)^{(3/2)}}}{1/(n)^{(5/4)}}$
$\lim\limits_{n \to \infty} \dfrac{8 (\ln n)}{(n)^{1/4}}=0$
Thus, the series converges by the limit comparison test.
