For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
We define a sequence of real numbers $(b_n)_{n \in \mathbb{N}}$ by setting $b_0 = 1$, $b_1 = -\frac{1}{2}$, then
$$\forall n \in \mathbb{N}^*, \quad b_{2n+1} = 0 \quad \text{and} \quad b_{2n} = \frac{(-1)^{n-1}(2n)! \zeta(2n)}{2^{2n-1}\pi^{2n}}.$$
Calculate $b_2$, $b_4$ and $b_6$ then $\zeta(2)$, $\zeta(4)$ and $\zeta(6)$.