Unendliche Summen (1)

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(1)   \begin{equation*} \sum \limits_{n = 1}^{\infty}{\frac{1}{n^{2}}} = \frac{\pi^2}{6} \end{equation*}

(2)   \begin{equation*}\sum \limits_{n = 1}^{\infty}{\frac{1}{n^{4}}} = \frac{\pi^4}{90} \end{equation*}

(3)   \begin{equation*}\sum \limits_{n = 1}^{\infty}{\frac{1}{n^{6}}} = \frac{\pi^6}{945} \end{equation*}

(4)   \begin{equation*}\sum \limits_{n = 1}^{\infty}{\frac{1}{n^{8}}} = \frac{\pi^8}{9450} \end{equation*}

(5)   \begin{equation*}\sum \limits_{n = 1}^{\infty}{\frac{1}{n^{10}}} = \frac{\pi^{10}}{93555} \end{equation*}

(6)   \begin{equation*}\sum \limits_{n = 1}^{\infty}{\frac{1}{n^{12}}} = \frac{691\pi^{12}}{638512875} \end{equation*}

(7)   \begin{equation*} \sum \limits_{n = 1}^{\infty}{\frac{(-1)^{n+1}}{(2n-1)^{3}}} = \frac{\pi^{3}}{32} \end{equation*}

(8)   \begin{equation*}\sum \limits_{n = 1}^{\infty}{\frac{(-1)^{n+1}}{(2n-1)^{5}}} = \frac{5\pi^{5}}{1536} \end{equation*}

(9)   \begin{equation*}\sum \limits_{n = 1}^{\infty}{\frac{1}{(2n-1)^{4}}} = \frac{\pi^{4}}{96} \end{equation*}

(10)   \begin{equation*}\sum \limits_{n = 1}^{\infty}{\frac{1}{(2n-1)^{6}}} = \frac{\pi^{6}}{960} \end{equation*}

Für p \in \mathbb{N}

(11)   \begin{equation*} \sum \limits_{n = 1}^{\infty}{\frac{1}{n^{2p}}} = { \frac{(-1)^{p+1}p\,\pi^{2p}}{(2p+1)!} - \sum \limits_{q = 1}^{p-1}\frac{(-1)^{p+q}\pi^{2p-2q}}{(2p-2q+1)!} \sum \limits_{n = 1}^{\infty}{\frac{1}{n^{2q}}}} \end{equation*}

(12)   \begin{equation*} \frac{p\,\pi^{2p}}{(2p+1)!} = \sum \limits_{q = 1}^{p}\frac{(-1)^{p+q}\pi^{2p-2q}}{(2p-2q+1)!} \sum \limits_{n = 1}^{\infty}{\frac{1}{n^{2q}}} \end{equation*}

(13)   \begin{equation*} \sum \limits_{n = 1}^{\infty}{\frac{1}{(2n-1)^{2p}}} = \frac{2^{2p-1}-1}{2^{2p}}\sum \limits_{n = 1}^{\infty}{\frac{1}{n^{2p}}} \end{equation*}

Für r\in \mathbb{R},\, r\ne{0}

(14)   \begin{equation*} \sum \limits_{n = 1}^{\infty}{\frac{1}{1+r^2n^{2}}} = \frac{1}{2} {\left[{\frac{\pi}{r}\coth{\left({\frac{\pi}{r}}\right)} - 1}\right]} \end{equation*}

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