Series (mathematics)

In mathematics, a series is the cumulative sum of a given sequence of terms. Typically, these terms are real or complex numbers, but much more generality is possible.

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

According to the number of terms, the series may be finite or infinite. The former is relatively easy to deal with. In fact, the finite series is identified with the sum of all terms and --apart of the elementary algebra-- there is no particular theory that applies. It turns out, however, 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. So it is sometimes tacitly understood, especially in the analysis, that the term "series" refers to the infinite series. In what follows we adopt this convention and concentrate on the theory of the infinite case.

Formal definition

Given a sequence $a_{1},a_{2},\dots$ of elements that can be added, let

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

Then, the series is defined as the sequence $\{S_{n}\}_{n=1}^{\infty }$ and denoted by $\sum _{n=1}^{\infty }a_{n}.$ For a single n, the sum $S_{n}$ is called the partial sum of the series.

If the sequence $(S_{n})$ has a finite limit, the series is said to be convergent. In this case we define the sum of the series as

$\sum _{n=1}^{\infty }a_{n}=\lim _{n\to \infty }S_{n}.$

Note that the sum (i.e. the numeric value of the above limit) and the series (i.e. the sequence $S_{n}$) are usually denoted by the same symbol. If the above limit does not exist - or is infinite - the series is said to be divergent.