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**Random Variable** In a proof of $\displaystyle \int_{-\infty}^{\infty} f(x) \ \frac{ \sin x}{x} \ dx = \int^{\pi}_{0} f(x) \ dx $ when $\displaystyle f(x) $ is $\displaystyle \pi $ - periodic on the entire real line, it is asserted that $\displaystyle \csc x = \sum_{k=-\infty}^{\infty} \frac{(-1)^{k}}{x + k \pi} $ .

Neither Maple nor Mathematica recognize this series, even for specific values of $\displaystyle x$. But numerically at least it appears to equal $\displaystyle \csc x $ for some specific values of $\displaystyle x$.

Any idea about the origins of this series?

EDIT: It's probably from a complex Fourier series.