if $\sum n! $ converges in $\mathbb{Z_p}$ ring then what about convergence of $\sum n$?

Question

I have read in some notebook about P-adic numbers , I accrossed that $\sum n! $ converges in $\mathbb{Z_p}$ but not's clear what value can take , Now my question here is : if $\sum n! $ converges in $\mathbb{Z_p}$ does $\sum n$ also converges in the same ring using comparaison creterion ?

Answer 1

Consider what convergence usually means in $\Bbb Z_p$: That more and more digits (counting from the right) stop changing as the sequence goes on. In base $p$ (for any prime $p$), the elements of the sequence $n!$ gets more and more trailing zeroes, meaning any specific digit of $\sum n!$ is eventually fixed and doesn't change any more, which again means that the sum converges. This doesn't happen with $\sum n$.

Be careful with the comparison test: the standard "ordering" of $p$-adic integers isn't directly related to the standard ordering of the integers: $1000$ is less than $100$, and $n!$ is usually much smaller than $n$.

Answer 2

Adjusting Stan Tendijck's comment to Jyrki Lahtonen's correct critique:

$\lvert n!\rvert_p \to 0$ for $n \to \infty$, so that's why it converges. The other one does not have that property (i.e. $\lvert n \rvert_p \not \to 0$ for $n \to \infty$.)