# Tag Archives: Bernoulli

## Zeta for Even integers proof (Bernoulli numbers)

$$\zeta(2k) \, = \, (-1)^{k-1} B_{2k} \frac{2^{2k-1}}{(2k)!}{\pi}^{2k}$$ $$\textit{proof}$$ We start by the product formula of the sine function $$\frac{\sin(z)}{z} = \prod_{n=1}^\infty \left(1-\frac{z^2}{n^2 \, \pi^2} \right)$$ Take the logarithm to both sides \log(\sin(z)) – \log(z) = \sum_{n=1}^\infty \log \left(1-\frac{z^2}{n^2 \, \pi^2} … Continue reading