Sequence

In mathematics, a sequence is a list of terms, written:

x₀, x₁, x₂, ...

The terms are all members of some set S; thus the above sequence is a sequence in S. For example, the following is a sequence of integers:

1, 2, 3, 4

A sequence may have a finite or infinite number of terms; thus, it is called either finite or infinite. Obviously, it is impossible to give all the terms of an infinite sequence. Infinite sequences are given by listing the first few terms, followed by an ellipsis.

Formally, a sequence can be defined as a function from N (the set of natural numbers) into some set S.

If S is the set of integers, then the sequence is an integer sequence.

If S is endowed with a topology then it is possible to talk about convergence of the sequence. This is discussed in detail in the article about limits.

For a given sequence the corresponding sequence of partial sums is called an infinite series.

E.g.: 1 + 1/2 + 1/4 + ... is a convergent series, meaning that the sequence 1, 1 + 1/2, 1 + 1/2 + 1/4, ... is convergent.

A subsequence is a sequence with some of its members omitted.

In biochemistry, a biopolymer's sequence is synonymous with its primary structure: the list of basic building blocks constituting the polymer. Determining such a sequence is called sequencing.

In mediæval Latin literature, a sequence (Latin sequentia) is a poem written in a non-classical metre that uses rhyme and an accentual (stress based) rather than quantititive (vowel length based) verse form.

See, for example: Pange Lingua; Dies Iræ

In music, a sequence is a passage which is successively repeated at different pitches.