I was playing with some graphs and I noticed that $\frac{1}{x}-\frac{1}{x+\pi}+\frac{1}{x+2\pi}$ looked similar to $\csc x$.
I then plugged in (to desmos.com) $\sum_{n=-500}^{500}\frac{(-1)^n}{x+nπ}$ and found that it was pretty much $\csc{x}$ on the interval near $(-500π,500π)$.
So, my question is, why is it that $\sum_{-\infty}^\infty \frac{(-1)^n}{x+nπ} = \csc x$
