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Say we have a series $a_0+a_1+a_2+...+a_n+...$ that has the partial sums $$S_0 = a_0$$ $$S_1 = a_0+a_1$$ $$...$$ $$S_n = a_0+a_1+...+a_n$$
The series converges if $\lim_{n\to\infty} S_n = S$, where S is the sum of the series, i.e. $S = \sum_{j=0}^\infty a_j$.
Why is the limit the sum of the series?
By definition, indeed.
There is no ground to speak of the "sum" of a series, say $1 - 1 + 1 - 1 + \cdots$, for one may argue that $(1-1) + (1-1) + \cdots = 0$ and another may argue that $1 - (1-1) - (1-1) - \cdots = 1$, a blatant contradiction.
To talk of the sum of an infinite series in a meaningful way, a covenant has been made, which is precisely the definition interesting you.
It's the definition of the "sum" of the series. It's a sensible definition in the sense that
If $S$ is the sum of the infinite sequence $a_n$, then as you sum more and more elements of $a_n$, the sum you get will be as close to $S$ as you want it to be.
That said, the "makes sense" is a justification, but not a "proof" that it is correct, since it is our decision to use that definition.
