Answer
converges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{1}{n \sqrt{n+\ln n}}$$
We use the limit comparison test with $ \sum \dfrac{1}{n^{3/2}}$:
\begin{align*}
\lim_{n\to \infty } \frac{a_n}{b_n} &=\lim_{n\to \infty } \frac{n^{3/2}}{n \sqrt{n+\ln n}} \\
&= \lim_{n\to \infty } \frac{1}{ \sqrt{1+\frac{\ln n}{n}}}\\
&=1
\end{align*}
Then the given series also converges.
