Unendliche Summen (3)

(1) $\begin{equation*} \sum\limits_{n=1}^{\infty}{\frac{1}{n(n+1)}}&= 1 \end{equation*}$

(2) $\begin{equation*}\sum\limits_{n=1}^{\infty}{\frac{p!}{n(n+1)(n+2)\cdots (n+p)}} &= \sum\limits_{l=1}^{p}{\frac{1}{l}\sum\limits_{j=0}^{l-1}{\binom{p}{j}(-1)^j}}, \quad p\in \mathbb{N} \end{equation*}$

(3) $\begin{equation*} \sum\limits_{n=p}^{\infty}{\frac{1}{\binom{n+1}{p+1}}} &= (p+1)\sum\limits_{l=1}^{p}{\sum\limits_{j=0}^{l-1}{\binom{p}{j}\frac{(-1)^j}{l}}}, \quad p\in \mathbb{N} \end{equation*}$

Unendliche Summen (3)

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