TITLE: What are significant differences between polynomial rings and the ring of formal power series?
QUESTION [6 upvotes]: I would like to know the difference between the polynomial ring and the ring of formal power series. We didn't mention it in our lectures so I have to understand it on my own. I saw some posts here and on the internet and I understand that:
Formal power series are a generalisation of polynomials, in that the coefficients can be infinite while in the polynomial they have to end at one point. Then I guess we can evaluate the polynomial and therefore have a function while this isn't so simple in power series since they do not always converge.
I feel like this is a difference between polynomials and power series, but what about them as rings? Does it make a difference? Are there other important differences?
Edit: I also saw on Wikipedia that there are differences when we consider topologies, but I haven't done any topological course until now. This is my very first algebra course.

REPLY [2 votes]: Even though this is not (by contemporary standards) an "elementary" explanation, I think it is much more functional and explanatory than most others.
(An example of a "bad explanation" is the idea that formal power series "are strings of symbols"... Seriously? Marks on the page?)
After my youthful dalliances with worse viewpoints, I seriously do think that the best way to describe "the formal power series ring $R[[x]]$" (for commutative ring $R$, surely with unit $1$) is as the projective limit of the quotients $R[x]/\langle x^n\rangle$. Whatever other structure the ring $R$ may have can be added to the "categorical" requirements.
So, yes, in any version of topology on $R$, in the formal power series ring $x$ is "very small", so small that any sum $\sum_{n\ge 0} r_n x^n$ converges, for any $r_n\in R$. But/and this pretense at metrization is not necessary, apart from an example of a construction, if/when we are content with the characterization of the formal power series ring as a projective limit.
(Namely, $R$-homs from another $R$-module to $R[[x]]$ are exactly compatible families of $R$-homs to the limitands $R[x]/\langle x^n\rangle$.)