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Plana’s summation formula for ∑^{∞}_{𝑚=1,3,⋯}𝑚⁻²sin(𝑚𝛼),𝑚⁻³cos(𝑚𝛼),𝑚⁻²𝐴^{𝑚},𝑚⁻³𝐴^{𝑚}

Plana’s summation formula for ∑^{∞}_{𝑚=1,3,⋯}𝑚⁻²sin(𝑚𝛼),𝑚⁻³cos(𝑚𝛼),𝑚⁻²𝐴^{𝑚},𝑚⁻³𝐴^{𝑚}

The four series <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row sigma-summation Underscript 0 Overscript normal infinity Endscripts sine left-parenthesis 2 k plus 1 right-parenthesis alpha slash left-parenthesis 2 k plus 1 right-parenthesis squared comma sigma-summation Underscript 0 Overscript normal infinity Endscripts cosine left-parenthesis 2 k plus 1 right-parenthesis alpha …