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Export date: Thu Sep 24 4:14:59 2020 / +0000 GMT

Endliche Summen(2)


begin{equation} sumlimits_{k=0}^{n}{leftlfloor {frac{k}{p}} rightrfloor} = frac{1}{2} {leftlfloor {frac{n}{p}} rightrfloor} {left({ 2,(n+1), - p{{leftlfloor {frac{n}{p}} rightrfloor} - p }}right)}, quad p in mathbb{N} end{equation}

begin{align} sumlimits_{k=0}^{n}{(-1)^kleftlfloor {frac{k}{p}} rightrfloor} =  &frac{1}{8} {left[ 2 left(1 + (-1)^p+2(-1)^n right) {leftlfloor {frac{n}{p}} rightrfloor} - {left( 1-(-1)^p right)} {left( 1-(-1)^{leftlfloor {frac{n}{p}} rightrfloor} right)} right]} \=  &frac{1}{2} (-1)^n {leftlfloor {frac{n}{p}} rightrfloor} - frac{1}{2} begin{cases} {-leftlfloor {frac{n}{p}} rightrfloor} &textrm{if } pequiv 0 pmod{2} \   {leftlfloor {frac{n}{p}} rightrfloor} bmod{2} &textrm{if } pequiv 1 pmod{2} end{cases} quad p in mathbb{N} end{align}

begin{align} sumlimits_{k=0}^{n}{kleftlfloor {frac{k}{p}} rightrfloor} &=  frac{1}{2}{leftlfloor {frac{n}{p}} rightrfloor} left[ {{n,(n+1)}, - frac{p}{6} {left({{leftlfloor {frac{n}{p}} rightrfloor} + 1} right)}  {left( {{2, pleftlfloor {frac{n}{p}} rightrfloor} + p,, - 3}right)}}right]  \ &=  frac{{leftlfloor {frac{n}{p}} rightrfloor} left[ {{6,n,(n+1)}, - {2,p^2}{leftlfloor {frac{n}{p}} rightrfloor}^2 } - 3,p,(p-1){leftlfloor {frac{n}{p}} rightrfloor} - p,(p-3)right]}{12} , quad p in mathbb{N} end{align}

begin{align} sumlimits_{k=0}^{n}{(-1)^k kleftlfloor {frac{k}{p}} rightrfloor} &= {leftlfloor {frac{n+1}{2}} rightrfloor} {leftlfloor {frac{n}{p}} rightrfloor} (-1)^n  +  sumlimits_{j=1}^{leftlfloor {frac{n}{p}} rightrfloor}{(-1)^{pj} leftlfloor {frac{pj}{2}} rightrfloor} \ notag &=  frac{2n+1}{4} {leftlfloor {frac{n}{p}} rightrfloor}(-1)^n + \ notag &quad + begin{cases} frac{p}{2} frac{{leftlfloor {frac{n}{p}} rightrfloor}left(leftlfloor {frac{n}{p}} rightrfloor + 1right)}{2} - frac{1}{4}{leftlfloor {frac{n}{p}} rightrfloor}   &textrm{if } pequiv 0 pmod{2} \   frac{p}{2} {leftlfloor frac{{leftlfloor {frac{n}{p}} rightrfloor} + 1}{2} rightrfloor} (-1)^{leftlfloor {frac{n}{p}} rightrfloor} + frac{1}{4} frac{1 - (-1)^{leftlfloor {frac{n}{p}} rightrfloor}}{2}   &textrm{if } pequiv 1 pmod{2} end{cases} quad p in mathbb{N} end{align}

begin{equation} {S_{n}}^{(r)}:=sumlimits_{k=1}^{n}{leftlfloor {k^{r}} rightrfloor} , , r in mathbb{R}, quad  rgt{0} end{equation}

begin{align} sumlimits_{k=1}^{n}{leftlfloor {sqrt{k}} rightrfloor} &= left( {n+1} right) leftlfloor {sqrt{n}}, rightrfloor - S_{leftlfloor {sqrt{n}}, rightrfloor}^{(2)} \ notag &= left( {n+1} right) leftlfloor {sqrt{n}}, rightrfloor - frac{{leftlfloor {sqrt{n}} rightrfloor left( {leftlfloor {sqrt{n}} rightrfloor+1} right) left( {2leftlfloor {sqrt{n}} rightrfloor+1} right)}}{6}end{align}

begin{align} sumlimits_{k=1}^{n}{leftlfloor {sqrt{k}} rightrfloor}^2 &= left( {n+1} right) {leftlfloor {sqrt{n}}, rightrfloor}^2 + S_{leftlfloor {sqrt{n}}, rightrfloor}^{(2)} - 2cdot S_{leftlfloor {sqrt{n}}, rightrfloor}^{(3)} \ notag &= left( {n+1} right) {leftlfloor {sqrt{n}}, rightrfloor}^2 + frac{{leftlfloor {sqrt{n}} rightrfloor left( {leftlfloor {sqrt{n}} rightrfloor+1} right) left( {2leftlfloor {sqrt{n}} rightrfloor+1} right)}}{6} \ notag &quad - frac{{{leftlfloor {sqrt{n}} rightrfloor}^2 {left( {leftlfloor {sqrt{n}} rightrfloor+1} right)}}^2}{2}  end{align}

begin{equation} sumlimits_{k=1}^{n}{left lfloor {k^frac{1}{3}} rightrfloor} = left( {n+1} right) {left lfloor {n^frac{1}{3}} rightrfloor} -frac{{left lfloor {n^frac{1}{3}} rightrfloor}^{2},{left( {{left lfloor {n^frac{1}{3}} rightrfloor} +1} right)}^2}{4}end{equation}

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

begin{equation} S_{n}^{left(frac{1}{p}right)}+{S}_{leftlfloor {n^frac{1}{p}} rightrfloor}^{(p)} = left( {n+1} right)cdot left lfloor {n^frac{1}{p}} rightrfloor, quad pinmathbb{R},, p ge {1} end{equation}

begin{align} sumlimits_{k=1}^{n}{kleftlfloor {{k^{frac{1}{p}}}} rightrfloor} = frac{1}{2} left( { n(n+1) {leftlfloor {n^{frac{1}{p}}} rightrfloor} + S_{leftlfloor {n^{frac{1}{p}}} rightrfloor}^{(p)} - S_{leftlfloor {n^{frac{1}{p}}}rightrfloor}^{(2p)} }right) end{align}

begin{align} sumlimits_{k=1}^{n}{(-1)^k,left lfloor {k^frac{1}{p}} rightrfloor} &= frac{1}{4} left( 2 ,(-1)^n {leftlfloor {n^frac{1}{p}} rightrfloor} + {(-1)^{leftlfloor {n^frac{1}{p}} rightrfloor} - 1} right) \ notag &= frac{1}{2} left( (-1)^n {leftlfloor {n^frac{1}{p}} rightrfloor} + {(-1)^{leftlfloor {n^frac{1}{p}} rightrfloor}}bmod {2} right), quad pinmathbb{R},, p ge {1}  end{align}

begin{equation}sumlimits_{k=1}^{n}{leftlfloor {log_2{(k)}} rightrfloor} = ({n+1}) leftlfloor {log_2{(n)}} rightrfloor - 2 left({2^{ leftlfloor {log_2{(n)}} rightrfloor} -1} right)end{equation}

begin{equation}sumlimits_{k=1}^{n}{leftlfloor {log{(k)}} rightrfloor} = ({n+1}) leftlfloor {log{(n)}} rightrfloor - frac{10}{9} left({10^{ leftlfloor {log{(n)}} rightrfloor} -1} right)end{equation}

begin{align}sumlimits_{{k=1}}^{n}{leftlfloor {log_p{(k)}} rightrfloor} &= ({n+1}) leftlfloor {log_p{(n)}} rightrfloor - frac{p}{p-1} left({p^{ leftlfloor {log_p{(n)}} rightrfloor} -1} right) \ notag &quad  pin mathbb{N},  , p gt 1 end{align}

begin{align}sumlimits_{{k=1}}^{n}{k,leftlfloor {log_p{(k)}} rightrfloor} &= frac{n(n+1)}{2} leftlfloor {log_p{(n)}} rightrfloor - frac{p}{2} frac{left({p^{ leftlfloor {log_p{(n)}} rightrfloor} -1} right) left({p^{ leftlfloor {log_p{(n)}} rightrfloor + 1} -1} right)}{p^2-1} \ notag &quad  pin mathbb{N},  , p gt 1 end{align}

begin{equation}sumlimits_{{k=1}}^{n}{(-1)^k,leftlfloor {log_p{(k)}} rightrfloor} = frac{(-1)^p+(-1)^n}{2} leftlfloor {log_p{(n)}} rightrfloor,  quad  pin mathbb{N},  , p gt 1 end{equation}
Post date: 2016-04-23 17:10:23
Post date GMT: 2016-04-23 15:10:23

Post modified date: 2016-05-13 01:33:56
Post modified date GMT: 2016-05-12 23:33:56

Export date: Thu Sep 24 4:14:59 2020 / +0000 GMT
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