p-adic series $\sum n!$

Question

Let $\mathbb{Z}_{11}$ the completion of $\mathbb{Z}$ with respect to the maximal ideal $(11)$. How I can check that the series $\sum_{n=0}^{+\infty}n!$ converges in $\mathbb{Z}_{11}$?

Answer 1

The $p$-adic metric is an ultrametric, meaning $|a+b|_p ≤ \max (|a|_p, |b|_p)$ for all $a,b ∈ ℚ_p$.

This tells you that a series $\sum_n a_n$ converges in $ℚ_p$ if and only if its general term sequence $(a_n)_n$ is a null sequence (not just only if as it is the case for the Euclidean metric).

So you only need to check whether $(n!)_n$ is a null sequence in the $11$-adic topology. Is it?

Answer 2

Since $|n!|_p\to 0$, it follows that the series converges. This follows from the general fact that $\sum_n a_n$ converges if $|a_n|_p\to 0$.