We have a sequence $(a_n) \in \mathbb{C}_p$ such that $p^n||a_n|| \rightarrow 0$. I have to prove that the series $$\sum_{n=0}^{\infty}a_n \frac{n!}{x(x+1)(x+2)\cdots(x+n)}$$ converges $\forall x \in \mathbb{C}_p \setminus \mathbb{Z}_p$
What can we say when $x \in \mathbb{Z}_p$?
This question is from an exercise in Koblitz's $p$ - adic book. It has a hint given in the back but I'm having trouble following and filling in the blanks, especially the lines where he asks to show that $p^N - p^{N-1}$ of the factors are of norm $1$, $p^{N-1} - p^{N-2}$ of the factors are of norm $1/p$ and so on. And the case when $n$ is not a power of $p$.
I'm still new at working with $p$ - adic power series and quite terrible at it, so if someone could help me out, that'll be great