We define a sequence $(a_n)_{n \geqslant 1}$ by setting
$$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$
We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. Justify that the function $S$ is of class $\mathcal{C}^\infty$ on $]-R, R[$ and, for every integer $n \in \mathbb{N}$, express $S^{(n)}(0)$ as a function of $n$.