FANDOM


$ \sum_{k=1}^\infty\frac{\cos(kx)}{k^2}=\frac{\pi^2}{6}-\frac{1}{2}\pi x+\frac{1}{4}x^2 $

$ \sum_{k=1}^\infty\frac{\sin(kx)}{k^3}=\frac{\pi^2}{6}x-\frac{1}{2}\pi x^2+\frac{1}{12}x^3 $

$ \sum_{k=1}^\infty\frac{{}_{2n}C_n}{4^n(2n+1)}=\frac{\pi}{2} $

$ 8G+\pi^2+\psi_1\left(\frac{1}{4}\right)=0 $

$ 8G+\psi_1\left(\frac{3}{4}\right)-6L_2\left(\frac{1}{6}\right)-6L_2\left(\frac{5}{6}\right)=0 $

$ \psi_1\left(\frac{5}{6}\right)-5\psi_1\left(\frac{2}{3}\right)+8L_2\left(\frac{1}{6}\right)-+8L_2\left(\frac{5}{6}\right)=0 $

$ \psi_1\left(\frac{1}{6}\right)+\psi_1\left(\frac{5}{6}\right)-4\pi^2=0 $

$ \psi_1\left(\frac{1}{4}\right)+\psi_1\left(\frac{3}{4}\right)-2\pi^2=0 $

$ 3\psi_1\left(\frac{1}{6}\right)-15\psi_1\left(\frac{1}{3}\right)+4\pi^2=0 $

$ -\psi_1\left(\frac{5}{6}\right)-\psi_1\left(\frac{1}{3}\right)+4\psi_1\left(\frac{2}{3}\right)=0 $

$ -32G+4\psi_1\left(\frac{1}{4}\right)-3\psi_1\left(\frac{1}{3}\right)-3\psi_1\left(\frac{2}{3}\right)=0 $

$ 3\pi^2-8G+\psi_1\left(\frac{1}{4}\right)-3\psi_1\left(\frac{1}{3}\right)-3\psi_1\left(\frac{2}{3}\right)=0 $

$ -2\psi_1\left(\frac{1}{4}\right)-2\psi_1\left(\frac{3}{4}\right)+\psi_1\left(\frac{5}{6}\right)+\psi_1\left(\frac{1}{6}\right)=0 $

$ -4\pi^3-\psi_2\left(\frac{1}{4}\right)+\psi_2\left(\frac{3}{4}\right)=0 $

$ -64\pi^3\sqrt{3}-9\psi_2\left(\frac{1}{6}\right)+63\psi_2\left(\frac{2}{3}\right)=0 $

$ -52\zeta(3)-9\psi_2\left(\frac{2}{3}\right)+\psi_2\left(\frac{5}{6}\right)=0 $

$ 8\pi^3\sqrt{3}+52\zeta(3)-54\psi_2\left(\frac{1}{3}\right)+\psi_2\left(\frac{1}{12}\right)+\psi_2\left(\frac{7}{12}\right)=0 $

$ -8\psi_2\left(\frac{1}{4}\right)+\psi_2\left(\frac{1}{8}\right)+\psi_2\left(\frac{5}{8}\right)=0 $