How can we make an assumption on something undefined?

Question

I have been learning generating functions lately, and I love the concept that when we represent a sequence in the form of a polynomial, the coefficient of $x$ has some meaning, the power of $x$ has some meaning, but $x$ itself does not carry any meaning.. the abstraction is beautiful. But here is where I have a doubt - say we have a generating function of the form -

$$1+x+x^2+x^3+\cdots$$

Now, in almost every text I have read, this generating function is written as -

$$1 + x^2+x^3+ \cdots = \sum_{n=0}^{\infty}x^n =\frac{1}{1-x}$$

I understand that the generating function here is actually a geometric series. And the summation of geometric series is -

$$\frac{a(r^n-1)}{r-1}$$ where $a$ is the first element and $r$ is the common ratio. Now, when $n \rightarrow \infty$, we can say -

$$\frac{a(r^{\infty}-1)}{r-1} \approx \frac{a}{1-r}$$ iff $r \in (0,1)$.

Now, my question is, in case of the generating function stated above, $r=x$ and we have already seen that $x$ itself is not defined. So, how can we write that - $$1+x+x^2+x^3+\cdots =\frac{1}{1-x}$$ without knowing anything about $x$ and obviously without making sure that it lies between $(0,1)$. How can we make assumptions on a variable that is itself undefined? and even if we did make an assumption, where did we prove that our assumption holds good?

Can someone please clarify this to me?.. thank you so much..

Answer 1

There are a few ways to resolve this issue formally.

(I) Simply say a generating function equals a given formula with the implicit assumption $x$ is in the domain of convergence, whatever that is.

(II) Think of a generating function $\sum f_n x^n$ as a cute shorthand for a function $f$ with domain $\mathbb{N}$. Then the sum and product of generating functions correspond to pointwise sum and convolution of the corresponding coefficient functions. Note different coefficient functions $f$, when turned into power series, can essentially be the same function of $x$ defined on the same domain, for instance $f_n=n!$ and $g_n=n!^2$ both define the constant function $1$ defined on the domain $\{0\}$.

(III) Think of infinite polynomials as elements of the formal power series ring $\mathbb{F}[[x]]$ with the so-called "$(x)$-adic topology." Here, $x$ does not represent an unknown number in the field $\mathbb{F}$ (whatever field you're using), but rather a new transcendental element. The topology is defined by the metric $d(f,g)=c^{-n}$ where $c>1$ is some constant and the first nonzero term in $f-g$ contains the power $x^n$. Thus, the more initial terms two formal power series have in common, the "closer" they are. In particular, higher powers of $x$ are closer and closer to $0$ in this topology (just like for actual numbers $x$ with $|x|<1$, but this is an artificial construction with $x$ not representing a number). Note all nonzero scalars are equally and maximally far away from being $0$ as possible (although this changes if you upgrade to formal Laurent series which may include negative powers of $x$).

In all three settings, the equation $1+x+x^2+\cdots=(1-x)^{-1}$ is to be interpreted as saying the element $1+x+x^2+\cdots$ is the multiplicative inverse of the element $1-x$.

Answer 2

From an algebraic standpoint, the manipulation of generating functions is nothing more than the algebra on formal power series.

A generating function is fully determined by the infinite sequence of its coefficients (and vice versa). For instance, notice in the treatment on Wikipedia [1] how addition and multiplication of formal series can be defined solely on the sequence of coefficients. Of course you can obtain those formulas by computing as with real or complex numbers with $x$ considered an "indeterminate number."

It's not an entirely trivial algebraic matter though. For instance the Cauchy product that defines multiplication of formal power series can be defined purely algebraically without considerations on convergence because each coefficient in the product of two series only depends on a finite number of coefficients of each factor. In other words, finite sums of finite products of real or complex numbers only require algebra, no analysis.

From that viewpoint, the answer to your question is that we are not making any assumption on $x$. For instance, ${1-x}$ is the inverse formal power series* to the formal power series $1+x+x^2+x^3+\cdots$, meaning that $(1+x+x^2+x^3+\cdots) (1-x) = 1$.

*A polynomial is a particular case of a formal power series, where coefficients past a certain degree are all zero.

[1] https://en.wikipedia.org/wiki/Formal_power_series#The_ring_of_formal_power_series