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In order to understand the concept of power series, I’d like to relate it precisely to the general notion of infinite series. My understanding of infinite series is as follows:

If $\{a_n\}$ is a sequence (say, a mapping from $\mathbb{N}$ to $\mathbb{R}$), and if $\{S_n\}$ is the sequence of partial sums based on $\{a_n\}$ (in such a way that the general term $S_n = a_1 + a_2 + \ldots + a_n$), and if, finally,

$$ \lim_{n \to \infty} S_n = L, $$

then the infinite series (informally, the infinite sum: $a_1 + a_2 + a_3 + \ldots$) is defined as

$$ \sum_{n=1}^{\infty} a_n = L. $$

My question is: given a power series, say $\sum_{n=1}^{\infty} x^n$, how can I find each of the elements of the above definition? I already do not manage to recover the sequence of partial sums. For example, it does not seem correct to say that $S_3 = x + x^2 + x^3$, because this is not a sum (a number) but an open expression (depending on $x$). Being unable to recover the sequence of partial sums, it goes without saying that I cannot recover the original sequence. I think that what misleads me is that there is a variable $x$ involved here, on top of the variable $n$ occurring in sequences. Hence the question: are power series functions of 2 variables? If not, since a power series is a function of $x$, what is the domain of this function and what is the target set? Is the target set a set of series (depending on variable $x$)?

Rócherz
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Vince Vickler
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    You can think of it (although this isn’t helpful, to me at least) as a function from $\mathbb{R}$ (or $\mathbb{C}$) to the space of formal series. To be a power series, means this map is of the following form $P(x)=\sum_{i=0}^\infty a_ix^i$ This is a formal series. Then we can talk about whether this series converges for specific $x$ values. You should really get used to this sort of thing. The Jacobian of a function for example is not a matrix, but a map from $\mathbb{R}^n$ to matrices. However, really, it is a matrix with functions as its entries. – Malady Apr 09 '25 at 19:12
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    Step by step: power series ( converged) is function of one variable. Partial sum for power series is function of two variables. Domain can be all $x$ for which series converged. – zkutch Apr 09 '25 at 19:13
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    A formal expression is still a sum. We can think of $x+x*+x^3$ as a function, but functions have to be defined with a domain, while $x+x^2+x^3$ as a function can be described on any ring. It could be on the integers, the rationals, the real numbers, the complex numbers, matrices. So we eventually realized we can, for any ring $R,$ we can define a ring of polynomials, $R[x],$ of all all polynomials with coefficient in $R.$ $x$ is just a formal variable, what you are calling an "expression." This ring has addition and multiplication, just like any other ring. – Thomas Andrews Apr 09 '25 at 19:34
  • @ThomasAndrews I like this better than my explanation. As I said, you can think of them as functions, but this is somehow not the right way. – Malady Apr 09 '25 at 21:59

2 Answers2

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If we're being formal, it's best to start with formal power series.

Let $R$ be a ring (commutative with $1$, say). The ring of formal power series over $R$ is denoted by $R[[x]]$, which as a set consists of $f \colon \mathbb{Z}_{\geq 0} \to R$ (i.e. infinite sequences $(f(0), f(1), \ldots)$ in $R$). No notion of convergence is imposed. Addition is done pointwise, and multiplication is given by convolution: $(fg)(n) = \sum_{i=0}^n f(i) g(n-i)$. Usually one writes $f$ as $f(0) + f(1) x + f(2) x^2 + \cdots$, in which case addition and multiplication correspond to naive algebraic identities.

If $R$ has additional structure, e.g. $R = \mathbb{R}$, you can upgrade a formal power series to one with additional structure. Your example of $\sum_{n=0}^\infty x^n$ is a perfectly well-defined element of $\mathbb{R}[[x]]$: it's the function $f(n) = 1$, or equivalently the sequence $(1,1,\ldots)$. Indeed, in the formal power series ring $R[[x]]$, we have the identity $\left(\sum_{n=0}^\infty x^n\right)\left(1-x\right) = 1$, again without needing any notion of convergence. But since we know of limits, we can associate a "radius of convergence $\rho_f$ about $0$" to any $f \in \mathbb{R}[[x]]$, which may be zero. Analysis tells us we have a well-defined map $e_f \colon (-\rho_f, \rho_f) \to \mathbb{R}$ given by $e_f(x_0) = \lim_{N \to \infty} \sum_{n=0}^N f(n) x_0^n$. More structure holds, e.g. $e_f$ is continuous on this open interval.

Formally you could consider the subring of $\mathbb{R}[[x]]$ with $\rho_f \geq M$ for some fixed $M$. Analysis tells us this is actually a subring. Then you can say we have a ring homomorphism $f \mapsto e_f$ from this subring to the ring of real continuous functions on $(-M, M)$ under pointwise addition and multiplication. In that sense, the formal power series identity $\left(\sum_{n=0}^\infty x^n\right)\left(1-x\right) = 1$ implies the analytic identity $\sum_{n=0}^\infty x^n = 1/(1-x)$ for $|x| < 1$.

The above is simply an illustration for how to make typical manipulations with infinite series rigorous. In practice, nobody writes down the map $e_f$ as a separate symbol, they simply trust that one can think of the expression $f(0) + f(1) x + f(2) x^2 + \cdots$ in an appropriately rigorous way whenever the need arises.

I'll say one shouldn't let oneself waste too much time with this level of formality. It's largely trivial symbol pushing.

Joshua P. Swanson
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The partial sums are functions of $x$. So you should write $$ S_3(x) = x + x^2 + x^3. $$ For each particular value of $x$ the sequence $S_n(x)$ (here $n$ is the increasing index) is a sequence of real numbers, which you analyze as in the question.

(Other answers and comments point toward a more formal way to discuss this issue. You can see some of that here: What actually is a polynomial? /)

Ethan Bolker
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