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Endliche Summen(4)


begin{align} sumlimits_{k=0}^{n}{p^k} &=frac{1 - p^{n+1}}{1-p} , quad pne{1} \ notag \  sumlimits_{k=0}^{n}{k, p^k} &=p,frac{1-left( {n+1} right){p^n}+n, p^{n+1}}{(1-p)^2} , quad pne{1} \ notag \ sumlimits_{i=0}^{n}{r^i}{s^{n-i}}&=frac{r^{n+1}-{s^{n+1}}}{r-s}, quad {r}ne{s} \ notag \ sumlimits_{i+j+k=n}{r^i,s^j,t^k} &=frac{r^{n+2}}{left( {r-s} right)left( {r-t} right)}+frac{s^{n+2}}{left( {s-r} right)left( {s-t} right)}+frac{t^{n+2}}{left( {t-s} right)left( {t-r} right)}, quad r lt s lt t  \ notag \ sumlimits_{k=0}^{n} {x^{left lfloor {frac{k}{p}} rightrfloor}} &= left({n, - p{left lfloor {frac{n}{p}} rightrfloor}+1}right),x^{left lfloor {frac{n}{p}} rightrfloor} + pfrac{x^{left lfloor {frac{n}{p}}rightrfloor} - 1}{x-1}, quad pne{0},xne{1} \ notag \ sumlimits_{k=0}^{n} {{left lfloor {frac{k}{p}} rightrfloor}x^k} &=  frac{left({1 - x^{pleft lfloor {frac{n}{p}} rightrfloor}}right)x^p -{(1-x^p)left lfloor {frac{n}{p}} rightrfloor}x^{n+1} } {(1-x)(1-x^p)}, quad pne{0},xne{1} \ notag \ sumlimits_{k=p}^{n} {frac{x^k}{left lfloor {frac{k}{p}} rightrfloor}} &=  frac{1}{1-x}{left[{x^p - frac{x^{n+1}}{left lfloor {frac{n}{p}} rightrfloor} - sumlimits_{j=2}^{left lfloor {frac{n}{p}} rightrfloor} {frac{x^{pj}}{j(j-1)}}  }right]}, quad pge{1},xne{1} \ notag \ sumlimits_{k=1}^{n} {x^{left lfloor {log_p(k)} rightrfloor}} &= (n+1){x^{left lfloor {log_p(n)} rightrfloor}} -1 + frac{p(1-x)}{1-px} {left({1 - {{px}^{left lfloor {log_p(n)} rightrfloor}}}right)} \ notag &quad p,xinmathbb{R},,pgt{1},,p,xlt{1}  \ notag \  sumlimits_{k=0}^{n} {x^{left lfloor {sqrt{k}} rightrfloor}} &= frac{1 + x} {(1-x)^2} +  left[n+1 - frac{ {left( {left lfloor {sqrt{n}} rightrfloor} x, - {left({{left lfloor {sqrt{n}} rightrfloor}+1}right)}right)}^2 + x }{(1-x)^2}right] {x^{left lfloor {sqrt{n}} rightrfloor}} \ notag  & quad xinmathbb{R},,xne{1}  \ notag \   cdots &= cdots  end{align}
Post date: 2016-04-23 17:11:00
Post date GMT: 2016-04-23 15:11:00

Post modified date: 2016-05-13 00:24:48
Post modified date GMT: 2016-05-12 22:24:48

Export date: Wed Aug 5 16:05:50 2020 / +0000 GMT
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