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Question Number 137111 by mnjuly1970 last updated on 29/Mar/21

Answered by Dwaipayan Shikari last updated on 29/Mar/21

Σ_(n=−∞) ^∞ (1/(n^2 +1))=1+2Σ_(n=1) ^∞ (1/(n^2 +1))=1+(2/2)(πcoth(π)−1)=π((e^(2π) +1)/(e^(2π) −1))

$$\underset{{n}=−\infty} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{1}}=\mathrm{1}+\mathrm{2}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{1}}=\mathrm{1}+\frac{\mathrm{2}}{\mathrm{2}}\left(\pi{coth}\left(\pi\right)−\mathrm{1}\right)=\pi\frac{{e}^{\mathrm{2}\pi} +\mathrm{1}}{{e}^{\mathrm{2}\pi} −\mathrm{1}} \\ $$

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