n-1-x-2n-2n-1-2-pix-2-cot-pix-

Question Number 138068 by Ñï= last updated on 09/Apr/21

\underset{n = 1}{\sum^{\infty}} x^{2 n} ζ (2 n) = \frac{1}{2} - \frac{π x}{2} \cot π x

Answered by Dwaipayan Shikari last updated on 09/Apr/21

\underset{n = 1}{\sum^{\infty}} \underset{k = 1}{\sum^{\infty}} \frac{x^{2 n}}{k^{2 n}} = \underset{k = 1}{\sum^{\infty}} \frac{x^{2}}{k^{2} - x^{2}} = \frac{x}{2} \underset{k = 1}{\sum^{\infty}} \frac{1}{k - x} - \frac{1}{k + x}

= \frac{x}{2} (ψ (x + 1) - ψ (1 - x)) = \frac{x}{2} (ψ (x) + \frac{1}{x} - ψ (1 - x))

= \frac{1}{2} - \frac{x π}{2} c o t (π x) ψ (x) - ψ (1 - x) = - π c o t (π x)