en.wikipedia.org/wiki/Series_(mathematics
You haven’t added anything other than reiterate that you are unfamiliar with the standard mathematical definition of the limit of a convergent series.
FWIW I’ll outline the logic.
(1) First we define the limit of a sequence a1, a2, a3, … by the “arbitrarily close” standard, which is formalized by the business about the epsilons that you may have seen. (I’ll skip writing the gory details unless requested).
(2) Then we would like to define what we mean by a notation like a1 + a2 + a3 + … The axioms for the real numbers only allow us to add up finitely many real numbers. So we have to DEFINE what an infinite sum is.
We form the sequence of partial sums: a1, a1 + a2, a 1 + a2 + a3, + … Each term is well-defined because it’s a finite sum of real numbers. Now if the resulting SEQUENCE of partial sums has a limit as given by (1), then we define the infinite sum as the limit of that sequence. It’s a definition.
That’s it. That’s really it.
Now if the question is whether that’s what mathematicians say, it is. Wikipedia agrees and so do hundreds of textbooks.
If the question is whether the mathematicians maybe got it philosophically wrong, that’s a different discussion; and one that I’m not entirely unsympathetic to. But you claim to dispute that this is how mathematicians define infinite sums, and you’re just wrong about that.