Endliche Summen(4)

\begin{align} \sum\limits_{k=0}^{n}{p^k} &=\frac{1 – p^{n+1}}{1-p} , \quad p\ne{1} \\ \notag \\  \sum\limits_{k=0}^{n}{k\, p^k} &=p\,\frac{1-\left( {n+1} \right){p^n}+n\, p^{n+1}}{(1-p)^2} , \quad p\ne{1} \\ \notag \\ \sum\limits_{i=0}^{n}{r^i}{s^{n-i}}&=\frac{r^{n+1}-{s^{n+1}}}{r-s}, \quad {r}\ne{s} \\ \notag \\ \sum\limits_{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 \\ \sum\limits_{k=0}^{n} {x^{\left \lfloor {\frac{k}{p}} \right\rfloor}} &= \left({n\, – p{\left \lfloor {\frac{n}{p}} \right\rfloor}+1}\right)\,x^{\left \lfloor {\frac{n}{p}} \right\rfloor} + p\frac{x^{\left \lfloor {\frac{n}{p}}\right\rfloor} – 1}{x-1}, \quad p\ne{0},x\ne{1} \\ \notag \\ \sum\limits_{k=0}^{n} {{\left \lfloor {\frac{k}{p}} \right\rfloor}x^k} &=  \frac{\left({1 – x^{p\left \lfloor {\frac{n}{p}} \right\rfloor}}\right)x^p -{(1-x^p)\left \lfloor {\frac{n}{p}} \right\rfloor}x^{n+1} } {(1-x)(1-x^p)}, \quad p\ne{0},x\ne{1} \\ \notag \\ \sum\limits_{k=p}^{n} {\frac{x^k}{\left \lfloor {\frac{k}{p}} \right\rfloor}} &=  \frac{1}{1-x}{\left[{x^p – \frac{x^{n+1}}{\left \lfloor {\frac{n}{p}} \right\rfloor} – \sum\limits_{j=2}^{\left \lfloor {\frac{n}{p}} \right\rfloor} {\frac{x^{pj}}{j(j-1)}}  }\right]}, \quad p\ge{1},x\ne{1} \\ \notag \\ \sum\limits_{k=1}^{n} {x^{\left \lfloor {\log_p(k)} \right\rfloor}} &= (n+1){x^{\left \lfloor {\log_p(n)} \right\rfloor}} -1 + \frac{p(1-x)}{1-px} {\left({1 – {{px}^{\left \lfloor {\log_p(n)} \right\rfloor}}}\right)} \\ \notag &\quad p,x\in\mathbb{R},\,p\gt{1},\,p\,x\lt{1}  \\ \notag \\  \sum\limits_{k=0}^{n} {x^{\left \lfloor {\sqrt{k}} \right\rfloor}} &= \frac{1 + x} {(1-x)^2} +  \left[n+1 – \frac{ {\left( {\left \lfloor {\sqrt{n}} \right\rfloor} x\, – {\left({{\left \lfloor {\sqrt{n}} \right\rfloor}+1}\right)}\right)}^2 + x }{(1-x)^2}\right] {x^{\left \lfloor {\sqrt{n}} \right\rfloor}} \\ \notag  & \quad x\in\mathbb{R},\,x\ne{1}  \\ \notag \\   \cdots &= \cdots  \end{align}

   

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