Formelsammlung Mathematik: Unendliche Reihen: Reihen zum Polylogarithmus

Zurück zu Unendliche Reihen

${\text{Li}}_2\left(\frac{1}{2}\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2{2}^k}=\frac{{\pi }^2}{12}-\frac{1}{2}{\log }^{2}2$

Blender3D: Vorlage:Klappbox

${\text{Li}}_3\left(\frac{1}{2}\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^3{2}^k}=\frac{7}{8}\zeta (3)-\frac{{\pi }^2}{12}\log 2+\frac{1}{6}{\log }^{3}2$

Blender3D: Vorlage:Klappbox

${\text{Li}}_2\left(\frac{1}{{\varphi }^2}\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2{\varphi }^{2k}}=\frac{{\pi }^2}{15}-{\log }^2\varphi$

${\text{Li}}_2\left(\frac{1}{\varphi }\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2{\varphi }^k}=\frac{{\pi }^2}{10}-{\log }^2\varphi$

${\text{Li}}_2(-\frac{1}{\varphi })=\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^k}{k^2{\varphi }^k}=-\frac{{\pi }^2}{15}+\frac{1}{2}{\log }^2\varphi$

Blender3D: Vorlage:Klappbox

${\text{Li}}_3\left(\frac{1}{{\varphi }^2}\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^3{\varphi }^{2k}}=\frac{4}{5}\zeta (3)+\frac{2}{3}{\log }^3\varphi -\frac{2{\pi }^2}{15}\log \varphi$