We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay (i.e., sequences $(\alpha_n)_{n \in \mathbb{N}}$ such that for every integer $k \in \mathbb{N}$, the sequence $(n^k \alpha_n)_{n \in \mathbb{N}}$ is bounded).
Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$ and $j \in \mathbb{N}$. We set, for $n \in \mathbb{N}$,
$$R_n(j) = \sum_{p=n+1}^{+\infty} p^j \alpha_p$$
Show that the sequence $(R_n(j))_{n \in \mathbb{N}}$ has rapid decay.