Question: Please suggest some **hint** to test the convergence of the following series
$$\sum\_{n=1}^\infty (-1)^n(\sqrt{n+1}-\sqrt n)$$

Answer: 
Hint: 

$$\sqrt{n+1}-\sqrt{n} = \frac{1}{\sqrt{n+1}+\sqrt{n}}$$
We have 
$$u\_n=(-1)^n(\sqrt{n+1}-\sqrt{n})=\frac{(-1)^n}{\sqrt{n+1}+\sqrt{n}}$$
So the sequence $(|u\_n|)\_n$ converges to $0$ and is monotone decreasing then by [Alternating series test](http://en.wikipedia.org/wiki/Alternating_series) the series $\sum\_n u\_n$ is convergent.