Answer
The series $\sum_{n=1}^{\infty} \frac{1}{n^{3/2} \ln n} $ converges.
Work Step by Step
We have the given series
$\sum_{n=1}^{\infty} \frac{1}{n^{3/2} \ln n} $
We apply the limit comparison test with $b_n=\frac{1}{n^{3/2}}$ (a convergent p-series with $p=3/2\gt 1$):
$L=\lim_{n\rightarrow\infty} \frac{a_n}{b_n}=\frac{n^{3/2}}{n^{3/2}\ln n}=0$
Thus, our starting series converges.
