Using the result of Q11, deduce that for all $\zeta \in \mathbb{U}$ and all $p \in \mathbb{Z}$,
$$\frac{\zeta^{p}}{\mathrm{e}^{\zeta} - 1} = \sum_{j=0}^{+\infty} (-1)^{j} \zeta^{j+p-1} \beta(\zeta)^{j}$$
where $\beta \in \mathcal{E}$ and $|\beta(\zeta)| \leqslant C < 1$ for all $\zeta \in \mathbb{U}$.