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?