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1-1-2-1-1-2-1-2-2-1-3-2-1-1-2-2-2-1-2-1-2-gt-0-Or-K-r-0-r-2-2-2-1-1-2-




Question Number 136645 by Dwaipayan Shikari last updated on 24/Mar/21
(1/(1+(η^2 /(1+(((η+1)^2 )/(1+(((η+2)^2 )/(1+(((η+3)^2 )/(1+..))  ))))))))=(1/2)ψ(((η+2)/2))−(1/2)ψ(((η+1)/2))  (η>0)  Or  K_(r=0) ^∞ (η+r)^2 =(2/(ψ((η/2)+1)−ψ(((η+1)/2))))
$$\frac{\mathrm{1}}{\mathrm{1}+\frac{\eta^{\mathrm{2}} }{\mathrm{1}+\frac{\left(\eta+\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{1}+\frac{\left(\eta+\mathrm{2}\right)^{\mathrm{2}} }{\mathrm{1}+\frac{\left(\eta+\mathrm{3}\right)^{\mathrm{2}} }{\mathrm{1}+..}\:\:}}}}=\frac{\mathrm{1}}{\mathrm{2}}\psi\left(\frac{\eta+\mathrm{2}}{\mathrm{2}}\right)−\frac{\mathrm{1}}{\mathrm{2}}\psi\left(\frac{\eta+\mathrm{1}}{\mathrm{2}}\right)\:\:\left(\eta>\mathrm{0}\right) \\ $$$${Or}\:\:\underset{{r}=\mathrm{0}} {\overset{\infty} {\boldsymbol{\mathrm{K}}}}\left(\eta+{r}\right)^{\mathrm{2}} =\frac{\mathrm{2}}{\psi\left(\frac{\eta}{\mathrm{2}}+\mathrm{1}\right)−\psi\left(\frac{\eta+\mathrm{1}}{\mathrm{2}}\right)} \\ $$

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