Let $\left(a_k\right)_{k \in \mathbb{N}}$ be a sequence of elements of $\mathbb{K}$. For every polynomial $p \in \mathbb{K}[X]$, show that the expression $$\sum_{k=0}^{+\infty} a_k D^k p$$ makes sense and defines a polynomial of $\mathbb{K}[X]$.
Let $\left(a_k\right)_{k \in \mathbb{N}}$ be a sequence of elements of $\mathbb{K}$. For every polynomial $p \in \mathbb{K}[X]$, show that the expression
$$\sum_{k=0}^{+\infty} a_k D^k p$$
makes sense and defines a polynomial of $\mathbb{K}[X]$.