Editing Sequence (section)

{{short description|Finite or infinite ordered list of elements}}
{{Redirect|Sequential}}
{{other uses}}
[[Image:Cauchy sequence illustration2.svg|right|thumb|350px|A part of an infinite sequence of [[real number]]s (in blue), indexed by a natural number <math display="inline">n</math>. This sequence is neither increasing, decreasing, convergent, nor [[Cauchy sequence|Cauchy]]. It is, however, bounded (by red dashed lines).]]
In [[mathematics]], a '''sequence''' is an enumerated collection of [[mathematical object|objects]] in which repetitions are allowed and [[order theory|order]] matters. Like a [[Set (mathematics)|set]], it contains [[Element (mathematics)|members]] (also called ''elements'', or ''terms''). The number of elements (possibly [[infinite number|infinite]]) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a [[function (mathematics)|function]] from [[natural number]]s (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an [[indexed family]], defined as a function from an ''arbitrary'' index set.

For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''[[finite set|finite]]'', as in these examples, or ''[[Infinite set|infinite]]'', such as the sequence of all [[even and odd numbers|even]] [[positive integer]]s (2, 4, 6, ...).

The position of an element in a sequence is its ''rank'' or ''index''; it is the natural number for which the element is the image. The first element has index 0 or 1, depending on the context or a specific convention. In [[mathematical analysis]], a sequence is often denoted by letters in the form of <math>a_n</math>, <math>b_n</math> and <math>c_n</math>, where the subscript ''n'' refers to the ''n''th element of the sequence; for example, the ''n''th element of the [[Fibonacci sequence]] ''<math>F</math>'' is generally denoted as ''<math>F_n</math>''.

In [[computing]] and [[computer science]], finite sequences are usually called ''[[string (computer science)|strings]]'', ''[[word (formal language theory)|words]]'' or ''[[list (computer science)|lists]],'' with the specific technical term chosen depending on the type of object the sequence enumerates and the different ways to represent the sequence in [[computer memory]]. Infinite sequences are called ''[[stream (computing)|streams]]''.

The empty sequence&nbsp;(&nbsp;) is included in most notions of sequence. It may be excluded depending on the context.