Does the following alternating series converge or diverge?

Question

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}$

Answer 1

Take the ratio: $$ \frac{a_{n+1}}{a_n} $$ and show that it is less than 1.

Answer 2

$$ \sum\limits_{n=1}^{\infty} \frac{\sin (n+1/2)\pi}{1 + \sqrt{n}}=\sum\limits_{n=1}^{\infty} \frac{(-1)^n}{1+\sqrt n} $$ 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 $$\sum _{n=0}^{\infty } \frac{(-1)^n}{\sqrt{n}+1}\approx 0.721717$$