calculus - Test for convergence/divergence of $sum_{n=1}^{infty}(-1)^nsinleft(frac{n}{pi}right)$

Given the series

$$\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$$

I need to test for convergence/divergence. I think the divergent test might work here. I could see that the $\lim_{n\rightarrow\infty}(-1)^n\sin(\frac{n}{\pi})$ might not exist, so the series is divergent. But I still need a solid proof here.

Any help is appreciated. Thanks.

Answer

I guess the standard argument should work. If $S_n=\sum_{k=0}^na_k$ converges then $a_k\rightarrow 0$. The necessary condition is not satisfied.