We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. We define the function $f$ on the segment $[-1,1]$ by:
$$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$
Show that $f$ is of class $C^\infty$ on $[-1,1]$.