Wikipedia states:
In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number. Thus a formal power series differs from a polynomial in that it may have infinitely many terms, and differs from a power series, whose variables can take on numerical values.
What I am getting from this is that in both polynomials and formal power series, the variables "don't represent numbers". But I'm not exactly sure what this means, or what they do represent. Also it seems to be inconsistent with how I've been using polynomials, which is very much as "variables representing numbers".
So basically I'm conceptually confused about what this means, and can't really understand how they're being used.
Best Answer
The best way to see this: forget about the $X$ in a polynomial or a formal power series, they are really sequences of coefficients, with no constraint on their values, with some specific rules of computation for addition and multiplication.
A power series, however, involves some limiting process, and that requires specific conditions, namely that the series converges.
For instance, you can manipulate $S=\sum_{n=0}^{\infty} x^n$ as a formal power series, and you won't consider convergence, only the operations on it, for instance $S^2=1+2x+3x^2+\dots$. That is, the coefficients of $S$ are $(1,1,1,\dots)$ while the coefficients of $S^2=(1,2,3,\dots)$. But you may also consider $T=\sum_{n=0}^{\infty} n! x^n$, it's a valid formal power series.
Now, for a power series, you require convergence. It's possible to prove that a power series in $x$ converges for all complex number $x$ such that $|x|<R$, for some real (of infinite) $R$. This $R$ is unique and is called the radius of convergence. For instance, the series $S$ above has radius $1$. It converges for $|x|<1$ to the number $\frac{1}{1-x}$. The series $T$ has radius $0$: it never converges if $x\ne0$. As a power series, it's almost useless, but as a formal power series, it can still be useful (we don't care that it does not converge).
There is a similar distinction between a polynom and a polynomial function. But here it's even more tricky, because in the usual undergraduate courses polynom are considered with coefficients in $\Bbb R$ or $\Bbb C$, and there many properties of polynomial directly relate to properties of the associated polynomial function.
When coefficients are in a finite field, it's more surprising. For instance, in $\Bbb F_2$, the finite field with two elements, the polynom $X^2+X$ is not the null polynomial (the null polynomial has null coefficients). However, the function $x\to x^2+x$ only takes the value $0$.