prove Σ_(r=−∞) ^∞ (1/(x + (r+(1/2))π)) = tan(x) (Σ_(r=−∞) ^∞ (1/(x + r)))(Σ_(r=−∞) ^∞ (1/(x + r))) = −(π^2 /4) ( r = odd) (r = even)


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Question Number 166168 by alcohol last updated on 14/Feb/22

$p r o v e$ $\underset{r = - \infty}{\sum^{\infty}} \frac{1}{x + (r + \frac{1}{2}) π} = t a n (x)$ $(\underset{r = - \infty}{\sum^{\infty}} \frac{1}{x + r}) (\underset{r = - \infty}{\sum^{\infty}} \frac{1}{x + r}) = - \frac{π^{2}}{4}$ $(r = o d d) (r = e v e n)$
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