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Endliche Summen(1)


begin{equation} sumlimits_{k=1}^{n}{k} = frac{nleft( {n+1} right)}{2} end{equation} begin{equation} sumlimits_{k=1}^{n}{k^2} = frac{nleft( {n+1} right)left( {2n+1} right)}{6} end{equation} begin{align} sumlimits_{k=1}^{n}{k^3} &= {left[ {frac{nleft( {n+1} right)}{2}} right]}^{2} \&= {left[ {sumlimits_{k=1}^{n}{k}} right]}^{2}\ &= sumlimits_{i=1}^{n}{sumlimits_{j=1}^{n}{ij}} end{align} begin{equation} sumlimits_{k=1}^{n}{k^4} =frac{nleft( {n+1} right)left( {2n+1} right)left( {3nleft( {n+1} right)-1} right)}{30} end{equation} begin{equation} sumlimits_{k=1}^{n}{k^5} =frac{{n^2}{left( {n+1} right)}^2 {left( {2nleft( {n+1} right)-1} right)}}{12}end{equation} begin{equation} sumlimits_{k=1}^{n}{k^6} =frac{nleft( {n+1} right),left( {2n+1} right),left[ {3nleft( {n+1} right),left[ {nleft( {n+1} right)-1} right]+1} right]}{42}end{equation} begin{equation} sumlimits_{k=1}^{n}{k^m}= frac{n^{m+1}}{m+1}+sumlimits_{j=1}^{m}{binom {m}{j} frac{(-1)^{j+1}}{j+1} sumlimits_{k=1}^{n}{k^{m-j}}} end{equation} begin{equation} {S_{n}}^{(m)}:=sumlimits_{k=0}^{n}{ {k^{m}} } , , m in mathbb{N_0} end{equation} begin{equation} sumlimits_{j=1}^{m}{binom {m}{j} (-1)^j S_{n}^{(m-j)}} = (-1)^m - n^{m}   end{equation} begin{equation} S_{n}^{(m)} = frac{n^{m+1}}{m+1} + sumlimits_{j=1}^{m}{binom {m}{j} frac{(-1)^{j+1}}{j+1} {S_{n}}^{(m-j)}} end{equation} begin{equation} S_{n}^{(m)} = frac{1}{m+1}{left[n^{m+1} + sumlimits_{j=1}^{m}{binom{m+1}{j+1} (-1)^{j+1} S_{n}^{(m-j)}}right]} end{equation} begin{equation} sumlimits_{j=1}^{m}{binom {m+1}{j} {S_{n}}^{(j)}} = {{left({n+1}right)}^{m+1} - n - 1}  end{equation} begin{equation} sumlimits_{j=0}^{m-1}{binom {m}{j} {S_{n}}^{(j)}} = {{left({n+1}right)}^{m} - delta_{m0}}  end{equation} begin{equation} {S_{n}}^{(m)} = frac{1}{m+1} left[{left({{left({n+1}right)}^{m+1} - n - 1}right)} - sumlimits_{j=1}^{m-1}{binom {m+1}{j} {S_{n}}^{(j)}}right] end{equation} begin{equation} sumlimits_{j=0}^{m}{binom {m-1}{j} 2^{m-j,}S_{n}^{(j)}} = {{left({n+2}right)}^{m} + {left({n+1}right)}^{m} - delta_{m0} - 1}  end{equation} begin{equation} {S_{n}}^{(m)} = sumlimits_{j=0}^{m}{binom{n+1}{j+1}j!,S(m,j)} end{equation} (S(m,j)) steht für das Stirling-Symbol zweiter Art. begin{equation} S(m,j) = frac{1}{j!}sum_{i=0}^{j}{(-1)^ibinom{j}{i}(j-i)^m} end{equation} begin{equation} {S_{n}}^{(m)} = sumlimits_{j=0}^{m} sum_{i=0}^{j} {binom{n+1}{j+1} {binom{j}{i}(-1)^i,(j-i)^m}} end{equation} begin{equation} {S_{n}}^{(m)} = (n+1)sumlimits_{j=1}^{m}{frac{(n)_j}{(j+1)}S(m,j)}  \ text{wobei } (n)_j = n(n-1)(n-2)cdots (n-j+1) end{equation} begin{equation} sumlimits_{k=1}^{n}{binom {n}{k} {k^m}} =  sumlimits_{j=0}^{m}{(n)_j, 2^{n-j}S(m,j)} \ text{wobei } (n)_j = n(n-1)(n-2)cdots (n-j+1) end{equation} begin{equation} sumlimits_{k=m}^{n}{(n-k)(k-m)} = frac{(n-m-1)(n-m)(n-m+1)}{6} end{equation}
Post date: 2016-04-23 17:04:32
Post date GMT: 2016-04-23 15:04:32

Post modified date: 2020-03-18 23:37:30
Post modified date GMT: 2020-03-18 22:37:30

Export date: Thu Sep 24 3:49:37 2020 / +0000 GMT
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