142 views

### How to compute $\sum_{k = 1}^{\infty} \frac{1}{k^2-\frac{1}{16}}$? [duplicate]

How do I compute $\sum_{k = 1}^{\infty} \frac{1}{k^2-\frac{1}{16}}$ ? Mathematica says the sum converges and it somewhat looks like the Basel problem, but so far I do not know how to approach it.
293 views

### Curious $\sum _{n=1}^{\infty} \frac{1}{n^2 - x^2}$ identity [duplicate]

Let $$F(x) = \sum _{n=1}^{\infty} \frac{1}{n^2 - x^2}$$ It seems that for odd integer $k$ $$F\left(\frac{k}{2}\right) = \frac{2}{k^2}$$ My evidence is strictly computational, and I have no idea how ...
105k views

### Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
6k views

3k views

### Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$

After numerical analysis it seems that $$\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$$ Could someone prove the validity of such identity?
### Method of proof of $\sum\limits_{n=1}^{\infty}\tfrac{\coth n\pi}{n^7}=\tfrac{19}{56700}\pi^7$
The following formula was stated by Ramanujan: $$\sum\limits_{n=1}^{\infty}\frac{\coth n\pi}{n^7}=\frac{19\pi^7}{56700}$$ Does anybody know the method of proof of this formula? I know that typically ...