Answer
The series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{ \sqrt{n^2+1}}$ converges.
Work Step by Step
We have the absolute series
$$\sum_{n=1}^{\infty} |\frac{(-1)^{n}}{ \sqrt{n^2+1}}|=\sum_{n=1}^{\infty} \frac{1}{ \sqrt{n^2+1}}$$
We have $$\lim_{n\to \infty }b_n=\lim_{n\to \infty }\frac{1}{ \sqrt{n^2+1}}=\lim_{n\to \infty }\frac{1/n}{\sqrt{1+1/n^2}=0}.$$
Since the terms are decreasing, positive, and tend to zero, then the series converges by the alternating series test.
