# Formelsammlung Mathematik: Unendliche Reihen: Reihen mit Integralkosinus

Zurück zu Unendliche Reihen

##### 1
${\displaystyle 2\cdot \sum _{n=1}^{\infty }{\text{Ci}}\left({\frac {n\pi }{2}}\right)=2\log 2-\gamma }$

##### 2
${\displaystyle 2\cdot \sum _{n=1}^{\infty }(-1)^{n}\,{\text{Ci}}\left({\frac {n\pi }{2}}\right)=-\gamma }$

##### 3
${\displaystyle 2\cdot \sum _{n=1}^{\infty }{\text{Ci}}(n\pi )=\log 2-\gamma }$

##### 4
${\displaystyle 2\cdot \sum _{n=1}^{\infty }(-1)^{n}\,{\text{Ci}}(n\pi )=1-\log 2-\gamma }$

##### 5
${\displaystyle 2\cdot \sum _{n=0}^{\infty }{\text{Ci}}\left(\left(n+{\frac {1}{2}}\right)\pi \right)=\log 2}$

##### 6
${\displaystyle 2\cdot \sum _{n=1}^{\infty }{\text{Ci}}(2n\pi )={\frac {1}{2}}-\gamma }$

##### 7
${\displaystyle 2\cdot \sum _{n=1}^{\infty }{\text{Ci}}((2n+1)\pi )=\log 2-{\frac {1}{2}}}$

##### 8
${\displaystyle 2\cdot \sum _{n=1}^{\infty }{\text{Ci}}(n\pi x)=\psi \left(1+\left\lfloor {\frac {x}{2}}\right\rfloor \right)-\log {\frac {x}{2}}}$

##### 9
${\displaystyle 2\cdot \sum _{n=1}^{\infty }(-1)^{n}\,{\text{Ci}}(n\pi x)=\psi \left({\frac {1}{2}}+\left\lfloor {\frac {x+1}{2}}\right\rfloor \right)-\log {\frac {x}{2}}}$

##### 10
${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k\pi }}\left({\frac {\pi }{2}}-{\text{Si}}(2\pi kx)\right)=x-\left(\lfloor x\rfloor +{\frac {1}{2}}\right)\log x+\log(\lfloor x\rfloor !)-\log {\sqrt {2\pi }}}$

##### 11
${\displaystyle {\frac {4}{\pi }}\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}\,{\text{Ci}}{\Big (}(2k+1)\pi x{\Big )}=(-1)^{\lfloor x+{\frac {1}{2}}\rfloor }\log(\pi x)-2\log \left({\sqrt {2\pi }}\,\,{\frac {\Gamma \left({\frac {3}{4}}\right)}{\Gamma \left({\frac {1}{4}}\right)}}\right)+2\sum _{n=0}^{\lfloor x-{\frac {1}{2}}\rfloor }(-1)^{n}\log \left((2n+1){\frac {\pi }{2}}\right)}$