If a series $\sum_{n = 1}^\infty a_n$ of real or complex numbers is convergent and the limit of its partial sums is $S$ then $S$ is the sum of the series. This circumstance may be denoted by $$\sum_{n = 1}^\infty a_n = S$$ or equivalently $$a_1+a_2+a_3+\cdots = S.$$ Nevertheless, one should not think that this means an addition of infinitely many numbers -- it's only a question of the limit $$\lim_{n\to\infty}\underbrace{(a_1+a_2+\cdots+a_n)}_{\textrm{partial sum}}.$$

The sum of the series is equal to the sum of a partial sum and the corresponding remainder term.