Convergence of $\sum_{n=1}^\infty (-1)^n(\sqrt{n+1}-\sqrt n)$
Solution 1:
Hint:
$$\sqrt{n+1}-\sqrt{n} = \frac{1}{\sqrt{n+1}+\sqrt{n}}$$
Solution 2:
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 the series $\sum_n u_n$ is convergent.