Answer
Convergent
Work Step by Step
Consider $a_n=\dfrac{(\ln n)^2}{n^{3/2}}$ and $b_n=\dfrac{1}{ n^{5/4}}$
Now, $\lim\limits_{n \to \infty}\dfrac{a_n}{b_n} =\lim\limits_{n \to \infty}\dfrac{\dfrac{(\ln n)^2}{n^{3/2}}}{\dfrac{1}{ n^{5/4}}}$
Thus, we have $ =\lim\limits_{n \to \infty} \dfrac{8 \ln n}{n^{1/4}}$
or, $=0$
Hence, the series converges due to the limit comparison test.
