Representations of p-adic integers as certain infinite sums 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?
 A: You want a map from ${\bf Z}[[x]]$ to that ring of coefficient-restricted power series, with kernel generated by $p-x$. Here's a start on constructing such. 
Any positive integer can be written in base $p$ as $a=a_0+a_1p+\cdots+a_rp^r$ with $0\le a_i\le p-1$ for each $i$. This gives you a map $\phi$ from positive integers to (coefficient-restricted) polynomials by $\phi(a)=a_0+a_1x+\cdots+a_rx^r$. Now all you have to do is extend the domain from the positive integers to ${\bf Z}[[x]]$. 
Start with the negative integers. In fact, start with $-1$; $$-1=(p-1)+(p-1)p+(p-1)p^2+\cdots$$ so $$\phi(-1)=(p-1)+(p-1)x+(p-1)x^2+\cdots$$ Now you can get $\phi(n)$ for any negative integer $n$ as a coefficient-restricted power series - details left to the reader. 
The more complicated problem is what to do after you've applied $\phi$ to each coefficient of an element of ${\bf Z}[[x]]$ and because of the interaction between coefficients you still have some (perhaps infinitely many) coefficients outside the desired range. Well, apply $\phi$ again, and again, and again. After $k$ applications, at least the first $k$ coefficients will be OK, and they will stay OK forever after, so in the limit, you have your isomorphism. 
There's probably a way of stating this in finite terms, but I'm not seeing it right now.  
A: The problem of showing that $\mathbb Z[[x]]/(x-p)$ coincides with the $p$-adic completion of $\mathbb Z$ has been addressed a couple of times already on this site: see here and here.
Recall that the $p$-adic completion of $\mathbb Z$ is the projective limit
$\mathbb Z_p := \varprojlim_n \mathbb Z/p^n$.
The isomorphism of $\mathbb Z[[x]]/(p-x)$ with $\mathbb Z_p$ is obtained as the projective limit of the maps
$\mathbb Z[[x]]/(x-p) \to \mathbb Z[[x]]/(x-p,x^n) = \mathbb Z/p^n$, the first
map being the natural surjection, and the second map being obtained by setting $x = p$.
The isomorphism of $\mathbb Z_p$ with the "power series" that you describe is
pretty straightforward: an element of $\mathbb Z/p^n$ has a unique representative in the form $a_0 + a_1 p + \cdots a_{n-1} p^{n-1}$ with
$0 \leq a_i < p$.  An element of $\mathbb Z_p$ thus can be described in a unique
way as a "power series" $\sum_{i \geq 0} a_i p^i$ with $0 \leq a_i < p$.
Composing the two indicated isomorphisms gives the isomorphism you asked about.
It is pretty explicit: given an element of $\mathbb Z[[x]]/(x-p)$, represented
by some power series $f(x)$, to get the first $n$ terms of the associated "power series", I take the degree $\leq n-1$ part of $f(x)$, replaced $x$ by $p$,
reduce modulo $p^n$, and then choose the coset representative of the form
$a_0 + \cdots a_{n-1} p^{n-1}$ as above.
