Zum Inhalt springen

Formelsammlung Mathematik: Unendliche Reihen: Reihen zum Polylogarithmus

Zurück zu Unendliche Reihen

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

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

${\displaystyle {\text{Li}}_{2}\left({\frac {1}{\phi ^{2}}}\right)=\sum _{k=1}^{\infty }{\frac {1}{k^{2}\,\phi ^{2k}}}={\frac {\pi ^{2}}{15}}-\log ^{2}\phi }$

${\displaystyle {\text{Li}}_{2}\left({\frac {1}{\phi }}\right)=\sum _{k=1}^{\infty }{\frac {1}{k^{2}\,\phi ^{k}}}={\frac {\pi ^{2}}{10}}-\log ^{2}\phi }$

${\displaystyle {\text{Li}}_{2}\left(-{\frac {1}{\phi }}\right)=\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k^{2}\,\phi ^{k}}}=-{\frac {\pi ^{2}}{15}}+{\frac {1}{2}}\log ^{2}\phi }$

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