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# Formelsammlung Mathematik: Unendliche Reihen: Lambertreihen

Zurück zu Unendliche Reihen

##### 1
${\displaystyle \sum _{n=1}^{\infty }{\frac {x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }\tau (n)\,x^{n}}$

##### 2
${\displaystyle \sum _{n=1}^{\infty }n^{a}\,{\frac {x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }\sigma _{a}(n)\,x^{n}}$

##### 3
${\displaystyle \sum _{n=1}^{\infty }\mu (n)\,{\frac {x^{n}}{1-x^{n}}}=x}$

##### 4
${\displaystyle \sum _{n=1}^{\infty }\varphi (n)\,{\frac {x^{n}}{1-x^{n}}}={\frac {x}{(1-x)^{2}}}}$

##### 5
${\displaystyle \sum _{n=1}^{\infty }\lambda (n)\,{\frac {x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }x^{n^{2}}}$

##### 6
${\displaystyle 1+4\sum _{n=1}^{\infty }\chi (n)\,{\frac {x^{n}}{1-x^{n}}}=\left(\sum _{n\in \mathbb {Z} }x^{n^{2}}\right)^{2}}$ mit ${\displaystyle \chi (n)=\left\{{\begin{matrix}0&{\text{falls}}&n\,\,\,{\text{gerade}}\\1&{\text{falls}}&n\equiv 1\,{\text{mod}}\,4\\-1&{\text{falls}}&n\equiv 3\,{\text{mod}}\,4\end{matrix}}\right.}$

##### 7
${\displaystyle 1+8\sum _{n=1}^{\infty }{\frac {n\,x^{n}}{1+(-x)^{n}}}=\left(\sum _{n\in \mathbb {Z} }x^{n^{2}}\right)^{4}}$

##### 8
${\displaystyle 1+16\sum _{n=1}^{\infty }{\frac {n^{3}\,x^{n}}{1-(-x)^{n}}}=\left(\sum _{n\in \mathbb {Z} }x^{n^{2}}\right)^{8}}$