# Formelsammlung Mathematik: Unendliche Reihen: Reihen mit zentrierten Binomialkoeffizienten

Zurück zu Unendliche Reihen

##### 1
${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{2k \choose k}}={\frac {2\pi {\sqrt {3}}+9}{27}}}$

##### 2
${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k\,{2k \choose k}}}={\frac {\pi }{3{\sqrt {3}}}}}$

##### 3
${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}\,{2k \choose k}}}={\frac {1}{3}}\zeta (2)}$

##### 4
${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{4}\,{2k \choose k}}}={\frac {17}{36}}\zeta (4)}$

##### 5
${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{2k \choose k}}={\frac {4{\sqrt {5}}\log \phi +5}{25}}}$

##### 6
${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k\,{2k \choose k}}}={\frac {2}{\sqrt {5}}}\log \phi }$

##### 7
${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k^{2}\,{2k \choose k}}}=2\log ^{2}\phi }$

##### 8
${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k^{3}\,{2k \choose k}}}={\frac {2}{5}}\zeta (3)}$

##### 9
${\displaystyle \sum _{k=0}^{\infty }{\frac {2^{2k-1}}{{2k \choose k}\,(2k+1)^{2}}}=G}$