Answer
The series converges by the Alternating Series test.
Work Step by Step
Let $b_n=\frac{1}{\sqrt{n}-1}$ for $n>2$. $b_n$ is always positive and decreasing. The $\lim\limits_{n\to\infty}b_n=0$. Thus the series $\sum_{n=2}^\infty\frac{(-1)^n}{\sqrt{n}-1}$ is convergent by the Alternating Series test.