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Question Number 166168 by alcohol last updated on 14/Feb/22
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)
$${prove} \\ $$$$\underset{{r}=−\infty} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{x}\:+\:\left({r}+\frac{\mathrm{1}}{\mathrm{2}}\right)\pi}\:=\:{tan}\left({x}\right) \\ $$$$\left(\underset{{r}=−\infty} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{x}\:+\:{r}}\right)\left(\underset{{r}=−\infty} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{x}\:+\:{r}}\right)\:=\:−\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:\:\:\left(\:{r}\:=\:{odd}\right)\:\:\:\:\:\:\:\:\left({r}\:=\:{even}\right) \\ $$

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