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How to relate precisely the definition of “power series” to the general definition of “infinite series” (based on sequences $\{a_n\}$ and $\{S_n\}$)

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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 $x$)?


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