Formelsammlung Mathematik: Unendliche Reihen: Reihen zur Psifunktion

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$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{\alpha +\beta k}=\frac{1}{2\beta }[\psi (\frac{1}{2}+\frac{\alpha }{2\beta })-\psi \left(\frac{\alpha }{2\beta }\right)]$

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$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{3k+1}=\frac{\log 2}{3}+\frac{\pi }{3\sqrt{3}}$

$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{4k+1}=\frac{\log (1+\sqrt{2})}{2\sqrt{2}}+\frac{\pi }{4\sqrt{2}}$

$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{5k+1}=\frac{\pi }{10}\csc \left(\frac{\pi }{5}\right)+\frac{1}{\sqrt{5}}\log \varphi +\frac{\log 2}{5}$