p-adic formal series, evaluations and invertible elements Take an invertible formal series $f\in \mathbb{Z}_p[[T]]$ of inverse $g\in \mathbb{Z}_p[[T]]$ and let $x\in \mathbb{Z}_p$ such that the value of $f$ evaluated at $x$ exists. I have a few questions:

*

*Is $f(x)$ a $p$-adic integer ? Since $\mathbb{Z}_p$ is a closed set I thought it was the case, but I have some doubts.

*Does $g(x)$ also exist ?

*If $g(x)$ does exist, is it the inverse of $f(x)$ ? I was thinking of evaluating at $x$ in the identity $f(T) g(T) =1$, but still not sure of myself.

I'm new to the p-adic world and to the formal series world, so thanks a lot.

You meant multiplicative inverse. Because the compositional inverse often exists in formal series.
If the series $f(x)$ converges then it does so to an element of $\Bbb{Z}_p$, yes.
Then try $f(T)=1-T$ and $x=1$.
If $g(x)$ converges as well then $f(x)g(x)=1$, yes (consider the truncated series, change the order of summation to make $\sum_n a_n b_{k-n}=0$ appear, show that the remainder is small).
