Series (mathematics)

Informally, series refers to the cumulative sum of a given sequence of terms. Typically, these terms are real or complex numbers, but much more generality is possible. The cumulative sum means that any series is a (special type of) sequence.

For example, for the sequence of the natural numbers 1,2,3,... the series is 1,1+2,1+2+3,...

More generally, given a finite or infinite sequence $a_{1},a_{2},...$ of elements that can be added, let

$S_{n}=a_{1}+a_{2}+\ldots +a_{n},\qquad n\in \mathbb {N}$

(to deal with the finite case assume that n is not bigger than the sequence). Then the series is defined as the sequence $\{S_{n}\}_{n=1}^{\infty }$ (with an obvious modification in the finite case). For a single n, the sum $S_{n}$ is called a partial sum of the series.