Formelsammlung Mathematik: Unendliche Reihen: Reihen mit harmonischen Zahlen

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$\sum _{n=1}^{\mathrm{\infty }}{H}_n{x}^n=-\frac{\log (1-x)}{1-x}|x|<1$

$\sum _{k=1}^{\mathrm{\infty }}\frac{H_k}{k^2}=2\zeta (3)$

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$\sum _{k=1}^{\mathrm{\infty }}\frac{H_k}{k^3}=\frac{{\pi }^4}{72}$

$2\sum _{k=1}^{\mathrm{\infty }}\frac{H_k}{k^n}=(n+2)\zeta (n+1)-\sum _{k=1}^{n-2}\zeta (n-k)\zeta (k+1)n\in {\mathbb{Z}}^{\ge 2}$

Blender3D: Vorlage:Klappbox

$2\sum _{k=1}^{\mathrm{\infty }}(-1{)}^{k-1}\frac{H_k}{k}=\frac{{\pi }^2}{6}-(\log 2{)}^2$

$2\sum _{k=1}^{\mathrm{\infty }}(-1{)}^{k-1}\frac{H_k}{k^{2n}}=(2n+1)\eta (2n+1)-2\sum _{k=0}^{n-1}\eta (2k)\zeta (2n+1-2k)n\in {\mathbb{Z}}^{\ge 1}$

$\sum _{k=1}^{\mathrm{\infty }}\frac{H_k}{k{2}^k}=\frac{{\pi }^2}{12}$

Blender3D: Vorlage:Klappbox

$\sum _{k=1}^{\mathrm{\infty }}\frac{H_k}{k^2{2}^k}=\zeta (3)-\frac{{\pi }^2}{12}\log 2$

$\sum _{k=1}^{\mathrm{\infty }}\frac{H_k^2}{k^2}=\frac{17{\pi }^4}{360}$

$\sum _{k=1}^{\mathrm{\infty }}\frac{H_k^2}{(k+1{)}^2}=\frac{11{\pi }^4}{360}$

$\sum _{k=1}^{\mathrm{\infty }}\frac{H_{k,r}}{k^n}=\left\{\begin{array}{c}\zeta (m)(\frac{1}{2}-\frac{(-1{)}^r}{2}(\genfrac{}{}{0ex}{}{m-1}{r})-\frac{(-1{)}^r}{2}(\genfrac{}{}{0ex}{}{m-1}{n}))\\ \\ +\frac{1-(-1{)}^r}{2}\zeta (n)\zeta (r)\\ \\ +(-1{)}^r\sum _{k=1}^{\lfloor r/2\rfloor }(\genfrac{}{}{0ex}{}{m-2k-1}{n-1})\zeta (2k)\zeta (m-2k)\\ \\ +(-1{)}^r\sum _{k=1}^{\lfloor n/2\rfloor }(\genfrac{}{}{0ex}{}{m-2k-1}{r-1})\zeta (2k)\zeta (m-2k)\end{array}\right\}n,r\in {\mathbb{Z}}^{\ge 2}$ und $m=n+r$ ungerade