Answer
The series $\sum_{n=1}^{\infty} \frac{ 1}{\sqrt{n^2+1}}$ diverges.
Work Step by Step
Use the limit comparison test with the p-series $\sum_{n=1}^{\infty} \frac{1}{n}$
Since we have
$$L=\lim_{n\to \infty } \frac{1/\sqrt{n^2+1}}{1/n }=\lim_{n\to \infty } \frac{ n }{\sqrt{n^2+1}}\\
=\lim_{n\to \infty } \frac{ 1}{\sqrt{1+1/n^2}}=1\gt0$$
then the series $\sum_{n=1}^{\infty} \frac{ 1}{\sqrt{n^2+1}}$ diverges because the p-series $\sum_{n=1}^{\infty} \frac{ 1}{ n }$, $p=1$, diverges.
