Unendliche Summen (2)

\[ \begin{align} \sum\limits_{n=1}^{\infty}\frac{\left\lfloor{\sqrt{n}}\right\rfloor}{n(n+1)} &= \frac{{\pi}^2}{6} \\ \notag \\  \sum\limits_{n=1}^{\infty}\frac{\left\lfloor{n^{\frac{1}{4}}}\right\rfloor}{n(n+1)} &= \frac{{\pi}^4}{90} \\ \notag \\  \sum\limits_{n=1}^{\infty}\frac{\left \lfloor {n^\frac{1}{p}} \right\rfloor}{n(n+1)} &= \sum\limits_{n=1}^{\infty} \frac{1}{{n}^{p}}, \quad {p}\in \mathbb{R}, \,p\gt{1}  \\ \notag \\  \sum\limits_{n=1}^{\infty}\frac{\left\lfloor{n^{\frac{1}{2}}}\right\rfloor}{n(n+1)(n+2)} &= \frac{3 + {\pi}^2\, -\, {3\,\pi} \coth \pi}{12} \\ \notag \\  \sum\limits_{n=1}^{\infty}\frac{\left \lfloor {n^\frac{1}{p}} \right\rfloor}{n(n+1)(n+2)} &= \frac{1}{2}\sum\limits_{n=1}^{\infty} \frac{1}{{n}^{p}{\left({{n}^{p} + 1}\right)}}, \quad {p}\in \mathbb{R}, \,p\gt{1}  \\ \notag \\  \sum\limits_{n=0}^{\infty} {{\left \lfloor {\frac{n}{p}} \right\rfloor}x^n} &=  \frac{x^p}{(1-x)\left(1-x^p\right)}, \quad p\in \mathbb{R}, p\ge{1},\mid{x}\mid\lt{1} \\ \notag \\ \sum\limits_{n=p}^{\infty} {\frac{x^n}{\left \lfloor {\frac{n}{p}} \right\rfloor}} &=  -\frac{1-x^p}{1-x}{\log\left(1-x^p\right)}, \quad p\in \mathbb{R}, p\ge{1},\mid{x}\mid\lt{1} \\ \notag \\  \sum\limits_{n=1}^{\infty}\frac{\left \lfloor {\sqrt{n}} \right\rfloor}{4n^{2}-1} &=  \frac{6 + {\pi}^2}{24}, \quad p\in\mathbb{R},\,p\gt{1}  \\ \notag \\      \sum\limits_{n=1}^{\infty}\frac{\left \lfloor {{n}^{\frac{1}{p}}} \right\rfloor}{4n^{2}-1} &=  \frac{1}{4} {\left( {1 + \sum\limits_{n=1}^{\infty} \frac{1}{{n}^{p}}}\right)}, \quad p\in\mathbb{R},\,p\gt{1}  \\ \notag \\        \sum\limits_{n=0}^{\infty} x^{\left \lfloor {\sqrt{n}} \right\rfloor} &= \frac{1+x}{(1-x)^2}, \quad x\in\mathbb{R},\, \mid x \mid \lt{1}  \\ \notag \\     \sum\limits_{n=1}^{\infty}\frac{\left \lfloor {\log_p(n)} \right\rfloor}{n(n+1)} &= \frac{1}{p-1}, \quad p\in\mathbb{R},\,p\gt{1}  \\ \notag \\ \sum\limits_{n=1}^{\infty}\frac{\left \lfloor {\log_p(n)} \right\rfloor}{4n^{2}-1} &=  \frac{1}{4\,(p-1)}, \quad p\in\mathbb{R},\,p\gt{1}  \\ \notag \\ \sum\limits_{n=1}^{\infty} x^{\left \lfloor {\log_p(n)} \right\rfloor} &= \frac{p-1}{1-p\,x}, \quad p,x\in\mathbb{R},\,p\gt{1},\,p\,x\lt{1}  \\ \notag \\ \sum\limits_{n=1}^{\infty} \frac{1}{p^{\left \lfloor {\log_{p-1}(n)} \right\rfloor}} &={p}\,{(p-2)}, \quad p\in\mathbb{R},\,p\gt{2}  \\ \notag \\ \sum\limits_{n=1}^{\infty} \frac{1}{p^{\left \lfloor {\log_{2}(n)} \right\rfloor}} &=\frac{p}{p-2}, \quad p\in\mathbb{R},\,p\gt{2}  \\ \notag \\ \cdots &= \cdots  \end{align}  \]

   

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