Does the following alternating series converge or diverge? I have the following series that I have to check for convergence or divergence:
$$
\sum\limits_{n=1}^{\infty} \frac{\sin (n+1/2)\pi}{1 + \sqrt{n}}
$$
I know that it is an alternating series therefore I have to check for two conditions to be satisfied in order for it to be convergent; the limit has to equal 0 as n approaches infinity and that for series $a_n$ $a_n < a_{n+1}$. I am able to prove that the limit is approaches zero but could someone help me prove that for the series above that:
$ a_n < a_{n+1}$

$$
\sum\limits_{n=1}^{\infty} \frac{\sin (n+1/2)\pi}{1 + \sqrt{n}}=\sum\limits_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt n}
$$
so this is an alternating series. Generic term converges to $0$ $$\underset{n\to \infty }{\text{lim}}\frac{1}{\sqrt{n}+1}=0$$
Therefore the series  converges and
$$\sum _{n=0}^{\infty } \frac{(-1)^n}{\sqrt{n}+1}\approx 0.721717$$
