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Question Number 122996 by Dwaipayan Shikari last updated on 21/Nov/20

Σ_(n=1) ^∞ (n/(n^4 +1))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{{n}^{\mathrm{4}} +\mathrm{1}} \\ $$

Commented by Dwaipayan Shikari last updated on 21/Nov/20

I have found  (i/2)(ψ(−(1/( (√2)))−(1/( (√2)))i)+ψ((−(1/( (√2)))+(1/( (√2)))i)−ψ((1/( (√2)))+(1/( (√2)))i)−ψ((1/( (√2)))−(1/( (√2)))i))

$${I}\:{have}\:{found} \\ $$$$\frac{{i}}{\mathrm{2}}\left(\psi\left(−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}{i}\right)+\psi\left(\left(−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}{i}\right)−\psi\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}{i}\right)−\psi\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}{i}\right)\right)\right. \\ $$

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