Let $p$ be a prime number. and $R[[X]]$ be the ring of formal series with coefficients in a $p$-adic field $R$. Let $\Lambda=\mathbb{Z}_p[[X]]$.

**Question 1)** Does there exist an explicit description of the integral closure $B$ of $\Lambda$ in $\mathbb{Q}_p[[X]]$ ?

Producing elements in $B$ but not in $\Lambda$ seems a little bit complicated. Any fraction $F(X)=P(X)/Q(X)$ with $P,Q \in \Lambda$, $X \nmid Q(X)$ can be seen as an element of $\mathbb{Q}_p[[X]]$ by expanding $Q(X)$ at 0. But actually, by Weierstrass preparation theorem, if $F \in B$ then $F$ is already in $\Lambda$. On the other hand, the $p$-adic exponential, defined by $$exp_p(X) = \sum_n \frac{1}{n!}X^n \in \mathbb{Q}_p[[X]]$$ is not in $\Lambda$, but is integral over $\Lambda$. Indeed, it is a $p$-th root of $exp_p(pX)$, which is in $\Lambda$, as the quantity $\frac{p^n}{n!}$ is $p$-integral for all integer $n \geq 0$.

I don't have any idea of what should $B$ look like, but as the set of coefficients of any element of $B$ satisfy polynomial relations with $\mathbb{Z}_p$-coefficients, my guess is that they should not grow too fast.

**Question 2)** Does a element of $B$ have a positive radius of convergence ? That is, if $F(X)=\sum_n c_n X^n \in B$, then does there exist an integer $r \geq 0$ such that the sequence $(p^{-rn}|c_n|_p)_n$ is bounded ?

Such an $F$ is an element in $\mathbb{Z}_p[[X/p^r]][\frac{1}{p}]$, and is even an element of $\mathbb{Z}_p[[X/p^r]]$, as $F$ is integral over $\mathbb{Z}_p[[X/p^r]]$. If it is true, it would reduce Question 1 to :

**Question 1')** What is the integral closure of $\Lambda$ in $\mathbb{Z}_p[[X/p^r]]$ ($r \geq 1$ fixed) ?