For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by
$$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
Show that for all $z \in \mathbb{C}$ such that $|z| < 2\pi$, we have
$$z = \left(e^z - 1\right)\left(1 - \frac{z}{2} + \sum_{k=1}^{+\infty} \frac{(-1)^{k-1}\zeta(2k)}{2^{2k-1}\pi^{2k}} z^{2k}\right)$$