I was messing around on Desmos (the online graphing calculator) the other day and accidentally created the following.
$\hskip{2.5in}$
This graph looks very similar to $\tan(x)$, which lead me to conjecture that
$$\tan(x)=\sum_{n=-\infty}^{\infty}\frac{-1}{x+\pi n + \frac{\pi}{2}}.$$
I went online searching for a proof and found nothing useful (mostly Taylor and Maclaurin series representations). Additionally, my personal attempts have been abysmal. Hence why I post this and request a proof. A reference to a proof would also be great, but I would still prefer it shown here.
Best Answer
The Mittag-Leffler theorem from complex variables in what you need. The only problem with your formula is that it's not clear how a doubly-infinite series should be evaluated. I presume you intend something like $$\tan{z}=\lim_{k\to\infty}\sum_{n=-k}^{k}\frac{-1}{x+\pi n + \frac{\pi}{2}}.$$ However, the usually convention is that both the "positive series" and the "negative series" must converge for the sum to be well-defined, and I don't think that's true in this case.
This is just a technical cavil, however. I'm really impressed that you found the series.
If you look at the formula for $\tan{z}$ in the link, you'll see the grouped the term for the pole at ${(2k+1)\pi\over2}$ with the term for the pole at ${-(2k+1)\pi\over2}$, to get an ordinary series.