It seems that $$\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi.$$ But I can't prove it. I cannot prove that the function is decreasing in $x$ either.
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asked Mar 31, 2022 at 21:31
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3$\begingroup$ Use the Poisson summation formula for both. $\endgroup$– fedjaCommented Mar 31, 2022 at 21:41
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16$\begingroup$ Isn't this also the Riemann sum for $\int_{\infty}^\infty dx/ (1 + x^2)$? $\endgroup$– David LoefflerCommented Mar 31, 2022 at 21:44
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$\begingroup$ Indeed. Bad question... Thank you both. $\endgroup$– Yaakov BaruchCommented Mar 31, 2022 at 21:48
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10$\begingroup$ More generally: $$\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2} = \pi\coth(\pi x).$$ $\endgroup$– Gerald EdgarCommented Apr 1, 2022 at 0:27
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1$\begingroup$ A related post on Mathematics: Formula for the series $f(x):=\sum\limits_{n=1}^\infty\frac{x}{x^2+n^2}$ $\endgroup$– Martin SleziakCommented Apr 1, 2022 at 3:54
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