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Endliche Summen(3)


begin{equation}sumlimits_{k=1}^{n}{frac{1}{kleft( {k+1} right)}} = frac{n}{n+1}end{equation}

begin{equation} sumlimits_{k=1}^{n}frac{left lfloor {k^frac{1}{p}} rightrfloor}{k(k+1)}= sumlimits_{k=1}^{leftlfloor {n^frac{1}{p}} rightrfloor}frac{1}{{k}^{p}} ,- frac{leftlfloor {n^frac{1}{p}} rightrfloor}{n+1}, quad pinmathbb{R},,pge{1}end{equation}

begin{equation} sumlimits_{k=1}^{n}frac{left lfloor {log_p(k)} rightrfloor}{k(k+1)}= frac{1 - p^{left lfloor {log_p(n)} rightrfloor}}{p-1} -frac{left lfloor {log_p(n)} rightrfloor}{n+1}, quad pinmathbb{R},,pgt{1}end{equation}

begin{equation} sumlimits_{k=1}^{n}frac{left lfloor {log_p(k)} rightrfloor}{4k^2-1}= frac{1 - p^{left lfloor {log_p(n)} rightrfloor}}{4(p-1)} -frac{left lfloor {log_p(n)} rightrfloor}{2(2n+1)}, quad pinmathbb{R},,pgt{1}end{equation}

begin{equation}sumlimits_{{k=1}}^{n}{{frac{1}{{kleft( {k+1} right)left( {k+2} right)}}}}=frac{{nleft( {n+3} right)}}{{4left( {n+1} right)left( {n+2} right)}}end{equation}

begin{align} frac{1}{p+1}sumlimits_{k=1}^{n}{frac{1}{binom{p+k}{p+1}}} &=sumlimits_{k=1}^{n}{frac{p,!}{k(k+1)(k+2)cdots (k+p)}} \ &= sumlimits_{l=1}^{p}{left({frac{1}{l}sumlimits_{j=0}^{l-1}{binom{p}{j}(-1)^j}} + {frac{1}{n+l}sumlimits_{j=l}^{p}{binom{p}{j}(-1)^j}}right)} , quad pin mathbb{N}, , n ge p \ &= sumlimits_{l=1}^{n}{frac{1}{l}sumlimits_{j=0}^{l-1}{binom{p}{j}(-1)^j}} + sumlimits_{l=1}^{p}{frac{1}{n+l}sumlimits_{j=l}^{n+l-1}{binom{p}{j}(-1)^j}} , quad pin mathbb{N}, , n lt p end{align}

begin{equation}sumlimits_{{k=1}}^{n}{{frac{1}{{left( {2k-1} right)left( {2k+1} right)}}}}=frac{n}{{2n+1}}end{equation}

begin{equation}sumlimits_{{k=1}}^{n}{{frac{1}{{left( {alpha left( {k-1} right)+beta } right)left( {alpha k+beta } right)}}}}=frac{n}{{beta left( {alpha n+beta } right)}}end{equation}

begin{align}sumlimits_{{1le {{i}_{1}}<{{i}_{2}}<{{i}_{3}}<cdots <{{i}_{{n-2}}}<{{i}_{{n-1}}}le n}} {frac{1}{{{i}_{1}}cdot {{i}_{2}}cdot {{i}_{3}} cdots {{i}_{{n-2}}}cdot {{i}_{n-1}}}}&=frac{1}{n,!}sumlimits_{1le {i}le n}{i}\&=frac{n+1}{2left( {n-1} right),!}end{align}
Post date: 2016-04-23 17:10:41
Post date GMT: 2016-04-23 15:10:41
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