Formelsammlung Mathematik: Unendliche Reihen: Reihen nach Ramanujan

Zurück zu Unendliche Reihen

1

$\sum _{k=1}^{\mathrm{\infty }}\frac{\coth (k\pi )}{(k\pi {)}^{4n-1}}=\sum _{k=0}^{2n}(-1{)}^{k-1}\frac{\zeta (2k)}{{\pi }^{2k}}\frac{\zeta (4n-2k)}{{\pi }^{4n-2k}}n\in {\mathbb{Z}}^{>0}$

2

$\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^k\text{csch}(k\pi )}{(k\pi {)}^{4n-1}}=\sum _{k=0}^{2n}(-1{)}^{k-1}\frac{\eta (2k)}{{\pi }^{2k}}\frac{\eta (4n-2k)}{{\pi }^{4n-2k}}n\in \mathbb{Z}$

3

$\sum _{k=0}^{\mathrm{\infty }}\frac{\tanh ((2k+1)\frac{\pi }{2})}{{((2k+1)\frac{\pi }{2})}^{4n-1}}=\sum _{k=0}^{2n}(-1{)}^{k-1}\frac{\lambda (2k)}{{\left(\frac{\pi }{2}\right)}^{2k}}\frac{\lambda (4n-2k)}{{\left(\frac{\pi }{2}\right)}^{4n-2k}}n\in {\mathbb{Z}}^{>0}$

4

$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k\text{sech}((2k+1)\frac{\pi }{2})}{{((2k+1)\frac{\pi }{2})}^{4n+1}}=\sum _{k=0}^{2n}(-1{)}^k\frac{\beta (2k+1)}{{\left(\frac{\pi }{2}\right)}^{2k+1}}\frac{\beta (4n-2k+1)}{{\left(\frac{\pi }{2}\right)}^{4n-2k+1}}n\in \mathbb{Z}$

5

$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k\text{sech}(\sqrt{3}(2k+1)\frac{\pi }{2})}{(2k+1)\frac{\pi }{2}}=\frac{1}{12}$

6

$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k\text{sech}(\frac{1}{\sqrt{3}}(2k+1)\frac{\pi }{2})}{(2k+1)\frac{\pi }{2}}=\frac{5}{12}$

7

$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k\text{sech}(\sqrt{3}(2k+1)\frac{\pi }{2})}{{((2k+1)\frac{\pi }{2})}^7}=\frac{1}{180}$

8

$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k\text{sech}(\frac{1}{\sqrt{3}}(2k+1)\frac{\pi }{2})}{{((2k+1)\frac{\pi }{2})}^7}=\frac{143}{27\cdot 180}$

9

$\sum _{k=0}^{\mathrm{\infty }}\frac{(2k+1{)}^{4n+1}}{e^{(2k+1)\pi }+1}=(2^{4n+1}-1)\frac{B_{4n+2}}{4(2n+1)}n\in {\mathbb{Z}}^{\ge 0}$

10

$\sum _{k=1}^{\mathrm{\infty }}\frac{k^{-4n+1}}{e^{2k\pi }-1}=-\frac{(2\pi {)}^{4n-1}}{4}\sum _{k=0}^{2n}(-1{)}^k\frac{B_{2k}}{(2k)!}\frac{B_{4n-2k}}{(4n-2k)!}-\frac{1}{2}\zeta (4n-1)n\in {\mathbb{Z}}^{\ge 0}$

11

$\sum _{k=1}^{\mathrm{\infty }}\frac{k^{4n+1}}{e^{2k\pi }-1}=\frac{B_{4n+2}}{4(2n+1)}n\in {\mathbb{Z}}^{>0}$

12

$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k(e^{2\pi k}-1)}=\frac{1}{4}\log \left(\frac{4}{\pi }\right)-\frac{\pi }{12}+\log \mathrm{\Gamma }\left(\frac{3}{4}\right)$

13

$\sum _{k=1}^{\mathrm{\infty }}\frac{k}{(-1{)}^k{e}^{\sqrt{3}\pi k}-1}=\frac{1}{24}-\frac{1}{4\sqrt{3}\pi }$

14

$\sum _{k\in \mathbb{Z}}\frac{1}{\cosh (k\pi )}=\frac{1}{\sqrt{2\pi }}\frac{\mathrm{\Gamma }\left(\frac{1}{4}\right)}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}$

15

$\sum _{k\in \mathbb{Z}}\frac{\pi }{{\cosh }^2((2k+1)\frac{\pi }{2})}=1$

16

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

17

$\sum _{k\in \mathbb{Z}}\frac{k\pi }{\sinh k\pi }=\frac{1}{2}+\frac{1}{8}\cdot \frac{{\mathrm{\Gamma }}^2\left(\frac{1}{4}\right)}{{\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}$

18

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

19

$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{\sinh (2^k\pi )}=\coth \pi -1$

20

$\sum _{k=0}^{\mathrm{\infty }}{(\genfrac{}{}{0ex}{}{-\frac{1}{2}}{k})}^3={\left(\frac{\mathrm{\Gamma }\left(\frac{9}{8}\right)}{\mathrm{\Gamma }\left(\frac{5}{4}\right)\mathrm{\Gamma }\left(\frac{7}{8}\right)}\right)}^2$

21

$\sum _{k=0}^{\mathrm{\infty }}(4k+1){(\genfrac{}{}{0ex}{}{-\frac{1}{2}}{k})}^3=\frac{2}{\pi }$

22

$\sum _{k=0}^{\mathrm{\infty }}(8k+1){(\genfrac{}{}{0ex}{}{-\frac{1}{4}}{k})}^4=\frac{2\sqrt{2}}{\mathrm{\Gamma }\left(\frac{1}{2}\right){\mathrm{\Gamma }}^2\left(\frac{3}{4}\right)}$

23

$\sum _{k=0}^{\mathrm{\infty }}(4k+1){(\genfrac{}{}{0ex}{}{-\frac{1}{2}}{k})}^5=\frac{2}{{\mathrm{\Gamma }}^4\left(\frac{3}{4}\right)}$