Answer
Diverges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^2+1}}$$
Since
\begin{align*}
\lim _{n \rightarrow \infty} \frac{n}{\sqrt{n^2+1}}&=\lim _{n \rightarrow \infty} \frac{1}{\sqrt{1+1/n^2}}\\
&=1 \neq 0
\end{align*}
Since the $n$th term $a_{n}$ does not converge to zero, thus the series diverges.
