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 length of the sequence). Then the series is defined as the sequence $\{S_{n}\}_{n=1}^{\infty }$ and denoted by $\Sigma _{n=1}^{\infty }a_{n}$ (with an obvious modification in the finite case). For a single n, the sum $S_{n}$ is called the partial sum of the series.

Finite series are relatively easy to understand and to deal with. It turns out that much care is required when manipulating infinite series. For example, some simple operations borrowed from elementary algebra -- as a change of order of the terms $a_{n}$ -- often lead to unexpected results. In what follows we concentrate on the theory of infinite series.