One way to define the p-adic integers is as the $p$-adic completion of $\mathbb{Z}$. With some additional work, it can be shown that this is isomorphic to $\mathbb{Z}[[x]]/(x-p)$.
Now, I know that another approach is to define $\mathbb{Z}_p$ as the ring of power series with powers of $p$ and coefficients from ${0, 1,..., p-1}$.
My question is: How can we see that the first definition of $\mathbb{Z}_p$ coincides with the second definition? and how can we find an explicit isomorphism from $\mathbb{Z}[[x]]/(x-p)$ to the ring of power series described above?