Question Number 203705 by SANOGO last updated on 26/Jan/24
$$\underset{{n}=\mathrm{0}} {\overset{+{oo}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} } \\ $$
Answered by witcher3 last updated on 26/Jan/24
$$\mathrm{decomposition} \\ $$$$\Sigma\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}+\mathrm{x}}=\frac{\mathrm{1}}{\mathrm{2}}\left(\Psi\left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\Psi\left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{2}}\right)\right) \\ $$$$\Sigma\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\left(\mathrm{n}+\mathrm{x}\right)^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{4}}\left(\Psi^{\mathrm{1}} \left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\Psi^{\mathrm{1}} \left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{2}}\right)\right) \\ $$