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Question Number 131580 by Dwaipayan Shikari last updated on 06/Feb/21

Prove or disprove Ī£_(n=0) ^āˆž (1/((n^2 +97)^2 ))=(š›‘^2 /(97(e^(š›‘(āˆš(97))) āˆ’e^(āˆ’š›‘(āˆš(97))) )^2 ))+(š›‘/(388)).((e^(2š›‘(āˆš(97))) +1)/(e^(2š›‘(āˆš(97))) āˆ’1))+((37635)/(37636))āˆ’(1/( 388(āˆš(97))))

$$\boldsymbol{\mathrm{Prove}}\:\boldsymbol{\mathrm{or}}\:\boldsymbol{\mathrm{disprove}} \\ $$$$\underset{\boldsymbol{{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\boldsymbol{{n}}^{\mathrm{2}} +\mathrm{97}\right)^{\mathrm{2}} }=\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{97}\left(\boldsymbol{{e}}^{\boldsymbol{\pi}\sqrt{\mathrm{97}}} āˆ’{e}^{āˆ’\boldsymbol{\pi}\sqrt{\mathrm{97}}} \right)^{\mathrm{2}} }+\frac{\boldsymbol{\pi}}{\mathrm{388}}.\frac{{e}^{\mathrm{2}\boldsymbol{\pi}\sqrt{\mathrm{97}}} +\mathrm{1}}{\boldsymbol{{e}}^{\mathrm{2}\boldsymbol{\pi}\sqrt{\mathrm{97}}} āˆ’\mathrm{1}}+\frac{\mathrm{37635}}{\mathrm{37636}}āˆ’\frac{\mathrm{1}}{\:\mathrm{388}\sqrt{\mathrm{97}}} \\ $$

Commented by Dwaipayan Shikari last updated on 06/Feb/21

I have this result but not sure..

$${I}\:{have}\:{this}\:{result}\:{but}\:{not}\:{sure}.. \\ $$

Commented by mindispower last updated on 09/Feb/21

seem exacte deduce this starting withe cot(x)=(1/x)+2Ī£_(nā‰„1) (x/(x^2 āˆ’Ļ€^2 n^2 )) coth(x)=āˆ’(1/x)āˆ’2Ī£(x/(x^2 +n^2 Ļ€^2 )) (1/y)coth(Ļ€y)=āˆ’(1/(Ļ€y^2 ))āˆ’(2/Ļ€)Ī£(1/(y^2 +n^2 )) Ī£(1/((y^2 +n^2 )))=(Ļ€/2)((1/(Ļ€y^2 ))āˆ’((coth(Ļ€y))/y)) ā‡’Ī£_(nā‰„1) ((2y)/((y^2 +n^2 )^2 ))=āˆ’(1/y^3 )+(Ļ€/(2y^2 ))coth(Ļ€y)āˆ’(Ļ€^2 /(2y))(āˆ’1+coth^2 (Ļ€y))

$${seem}\:{exacte}\: \\ $$$${deduce}\:{this} \\ $$$${starting} \\ $$$${withe}\: \\ $$$${cot}\left({x}\right)=\frac{\mathrm{1}}{{x}}+\mathrm{2}\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{{x}}{{x}^{\mathrm{2}} āˆ’\pi^{\mathrm{2}} {n}^{\mathrm{2}} } \\ $$$${coth}\left({x}\right)=āˆ’\frac{\mathrm{1}}{{x}}āˆ’\mathrm{2}\Sigma\frac{{x}}{{x}^{\mathrm{2}} +{n}^{\mathrm{2}} \pi^{\mathrm{2}} } \\ $$$$\frac{\mathrm{1}}{{y}}{coth}\left(\pi{y}\right)=āˆ’\frac{\mathrm{1}}{\pi{y}^{\mathrm{2}} }āˆ’\frac{\mathrm{2}}{\pi}\Sigma\frac{\mathrm{1}}{{y}^{\mathrm{2}} +{n}^{\mathrm{2}} } \\ $$$$\Sigma\frac{\mathrm{1}}{\left({y}^{\mathrm{2}} +{n}^{\mathrm{2}} \right)}=\frac{\pi}{\mathrm{2}}\left(\frac{\mathrm{1}}{\pi{y}^{\mathrm{2}} }āˆ’\frac{{coth}\left(\pi{y}\right)}{{y}}\right) \\ $$$$\Rightarrow\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{2}{y}}{\left({y}^{\mathrm{2}} +{n}^{\mathrm{2}} \right)^{\mathrm{2}} }=āˆ’\frac{\mathrm{1}}{{y}^{\mathrm{3}} }+\frac{\pi}{\mathrm{2}{y}^{\mathrm{2}} }{coth}\left(\pi{y}\right)āˆ’\frac{\pi^{\mathrm{2}} }{\mathrm{2}{y}}\left(āˆ’\mathrm{1}+{coth}^{\mathrm{2}} \left(\pi{y}\right)\right) \\ $$

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