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A real series $S_n$ is the limit to infinity of the sequence of partial sums of a real sequence $\sequence {a_n}$:

\(\ds S_n\) \(=\) \(\ds \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty a_n\)
\(\ds \) \(=\) \(\ds a_1 + a_2 + a_3 + \cdots\)

Historical Note

Much of the original work on series of real and complex numbers was done by Leonhard Paul Euler.

The main bulk of the work to placed the concept on a rigorous footing was done by Carl Friedrich Gauss, Niels Henrik Abel‎ and Augustin Louis Cauchy.