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prove-that-pi-4cos-pi-2-p-0-2p-1-1-p-2p-1-2-2-R-Z-




Question Number 28818 by abdo imad last updated on 30/Jan/18
prove that  (π/(4cos(((πα)/2))))=Σ_(p=0) ^∞     (((2p+1)(−1)^p )/((2p+1)^2  −α^2 ))  α ∈R−Z.
$${prove}\:{that}\:\:\frac{\pi}{\mathrm{4}{cos}\left(\frac{\pi\alpha}{\mathrm{2}}\right)}=\sum_{{p}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{\left(\mathrm{2}{p}+\mathrm{1}\right)\left(−\mathrm{1}\right)^{{p}} }{\left(\mathrm{2}{p}+\mathrm{1}\right)^{\mathrm{2}} \:−\alpha^{\mathrm{2}} } \\ $$$$\alpha\:\in{R}−{Z}. \\ $$

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