$p$-adic completion of $\mathbb{Z}[X]$ and $\mathbb{Z}[[X]]$. Let $p$ be prime. The $p$-adic completion of $\mathbb{Z}$ is the ring $\mathbb{Z}_p$ of $p$-adic integers, and its elements can be thought of as power series in $p$. Is there a nice description of the elements of the $p$-adic completions of $\mathbb{Z}[X]$ and $\mathbb{Z}[[X]]$?
Edit: The completion of $\mathbb{Z}[X]$ contains $\mathbb{Z}_p[X]$ of course. I think it should also contain the series $\sum_{i=0}^\infty p^iX^i$, so the completion is strictly bigger than $\mathbb{Z}_p[X]$. 

The $p$-adic completion of $\mathbf Z[[X]]$ is $\mathbf Z_p[[X]]$. To prove it, it suffices to prove that the map $\mathbf Z[[X]] \to\mathbf Z_p[[X]]$ induces an isomorphism on the $p$-adic completions. This is obvious because $$\mathbf Z[[X]]/p^n\mathbf Z[[X]] \to\mathbf Z_p[[X]]/p^n\mathbf Z_p[[X]]$$
is an isomorphism, both sides identifying with $(\mathbf Z/p^n\mathbf Z)[[X]]$ in a natural way.
However, the $p$-adic completion of $\mathbf Z[X]$ is the ring $\mathbf Z_p\left<X\right>$, the subring of $\mathbf Z_p[[X]]$ consisting of the formal power series $\sum_{n\geq 0} a_n T^n$ such that $|a_n| \to 0$ in $\mathbf Z_p$. It is a nice exercise in limits to prove it.
Such rings are known as Tate algebras and their study forms the starting point of rigid analytic geometry (which is to algebraic geometry over $p$-adic fields what complex analytic geometry is to algebraic geometry over $\mathbf C$).
