I know that the definition of an infinite series is the limit as $n \to \infty$ of its partial sums.
$$\underbrace{\sum_{n=0}^{\infty} \ \ a_n}_\text{Infinite Series}\ \stackrel{\text{def}}{=} \ \lim_{n \to \infty} \ \underbrace{\sum_{i=0}^{n} \ a_i}_\text{Partial Sums}$$
But then how do you define a finite series?
If a finite series is defined like I've done below, is that not recursive, as you're essentially defining something in terms of itself, as the partial sum of a finite series, is itself a finite series?
$$\underbrace{\sum_{i=0}^{n} \ \ a_i}_\text{Finite Series}\ \stackrel{\text{def}}{=} \ \lim_{\gamma\ \to\ n} \ \underbrace{\sum_{i=0}^{\gamma} \ a_i}_\text{Partial Sums}$$
At some point we have to define these series as the sum of terms of a infinite sequence $\{ a_n \} _{n=1}^{\infty}$, for the infinite series (where we view the partial sums as the finite sums of terms of this infinite sequence), and as the sum of terms of a finite sequence $\{ a_i \} _{i=1}^{n}$ for the finite series (and I'm not sure how to view the partial sums here, perhaps as finite sums of terms of the finite sequence where $i \leq n$). I'm not sure how to do this.
In short I'm having trouble with the definition of a finite series, and I'm having trouble making the connection between finite sequences and the definition of finite series, and how the two (sequences and series) relate to each other.
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$\begingroup$A finite series is just a sum: $$\sum_{j=1}^na_j=a_1+a_2+\dots+a_n.$$
Note If you actually didn't know what $\sum_{j=1}^n a_j$ meant you should stop reading here! The rest of this answer is likely to just cause more confusion. Otoh if you're wondering how one "formalizes" that definition then read on:
Of course an official formal definition cannot contain ellipses (...); the definition above depends on the reader recognizing a certain pattern, and the formal axioms and definitions should not depend on anything that fuzzy. When you see a definition involving "..." that's typically shorthand for a recursive definition. Formally, one defines $\sum_{j=1}^n a_j$ like so: $$\sum_{j=1}^1 a_j=a_1,$$ $$\sum_{j=1}^{n+1}a_j=a_{n+1}+\sum_{j=1}^n a_j.$$
$\endgroup$ $\begingroup$A finite series is just adding together a bunch of things:
$$\sum_{i=1}^na_i=a_1+a_2+\cdots+a_n . $$ This definition is unambiguous. There is also no need for partial sums because we do not need to (nor can we) describe the sum in terms of a limit.
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