Polynomial vs power series vs formal power series?

Question

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.

Answer 1

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).

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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 $0$.

Answer 2

The Wikipedia definitions are not perfect. Let's go at this a little differently.

A formal polynomial is a finite ordered list of coefficients. We use powers of an indeterminant to mark where in the list each coefficient is. For instance (where I write out all the numbers that are normally elided) $1 x^2 + 2 x^1 + 3 x^0$ is a different polynomial from $3 x^2 + 2 x^1 + 1 x^0$ because, for instance, the coefficients in the $x^2$ place are different. Notee the similarity between this notion and positional notation for writing numbers.

A formal power series is a potentially infinite ordered list of coefficients. We use powers of an indeterminant to mark where in the list each coefficient is. Note that formal polynomials are formal power series (where, once the power of the marking is high enough, all the coefficients are zero).

Notice that for formal objects, the indeterminate is just a marking tool, it is not a "variable" or something that can be "evaluated". We can add these ideas. We can evaluate these objects by specializing the indeterminate to a value. When you do this, you get polynomials and power series.

For power series, a new idea comes into play: convergence. Some lists of coefficients and certain specializations of the indeterminate may not produce an infinite sum that has a value. An easy example is specializing the formal power series with all coefficients set to $1$ by setting the indeterminant also to $1$, so the power series is a sum of infinitely many copies of $1$, which does not converge. So, while a formal power series has no convergence issues (because you never pretend to evaluate a formal power series), a power series has issues of convergence.

Notice that because we are writing (finite or infinite) lists as formal power objects, we can add them (just like in positional notation), subtract, and multiply them, which operations should seem familiar. We can also interleave them, reverse them, concatenate them, sort them, and do other operations that are sensible for lists, which perhaps seems less familiar from the context of polynomials.