Q:

convergence of $\sum_{n}^{\infty}(-1)^n \log(1+\sin(\sqrt{n+1}-\sqrt n)$

I found that series $\sum_{n}^{\infty}(-1)^n \log(1+\sin(\sqrt{n+1}-\sqrt n)$ is not absolutely convergent, but $\lim a_n = 0$, so I suspect that this series could converge, I want to use Leibniz test, but I don't know how to prove than the sequence is not increasing. How do I do that? 

A:

$\ln(1+x)$ is increasing for positive $x$, $sin(x)$ is increasing on $[0,\pi/2]$ so all you need to show is that the sequence $(\sqrt{n+1} - \sqrt{n})$ is non increasing (and takes values in $[0, \pi/2]$). do you know how to do that?

