Sequence

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For other senses of this word, see sequence (disambiguation).

In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and the exact same elements can appear multiple times at different positions in the sequence.

For example, (C,R,Y) is a sequence of letters that differs from (Y,C,R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2,4,6,...).

1 Examples and notation
2 Types and properties of sequences
3 Sequences in analysis
4 Series
5 Infinite sequences in theoretical computer science
6 Sequences as vectors
7 Doubly-infinite sequences
8 See also
9 External links

[edit] Examples and notation

There are various and quite different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the notations introduced below.

A sequence may be denoted (a₁,a₂, ...). For shortness, the notation (a_n) is also used.

A more formal definition of a finite sequence with terms in a set S is a function from {1,2,...,n} to S for some n≥0. An infinite sequence in S is a function from {1,2,...} (the set of natural numbers) to S.

A finite sequence is also called an n-tuple. A function from all integers into a set is sometimes called a bi-infinite sequence, since it may be thought of as a sequence indexed by negative integers grafted onto a sequence indexed by positive integers.

Finite sequences include the null sequence ( ) that has no elements.

[edit] Types and properties of sequences

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.

If the terms of the sequence are a subset of an ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This is a special case of the more general notion of monotonic function.

The terms non-decreasing and non-increasing avoid any possible confusion with strictly increasing and strictly decreasing, respectively, see also strict.

If the terms of a sequence are integers, then the sequence is an integer sequence. If the terms of a sequence are polynomials, then the sequence is a polynomial sequence.

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

[edit] Sequences in analysis

In mathematical analysis, when talking about sequences, one usually understands sequences of the form

(x 1, x 2, x 3,...)

(x 0, x 1, x 2,...)

i.e. infinite sequences of elements indexed by natural numbers. (It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by x_n = 1/log(n) would be defined only for $n\ge2$ . When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given $N$ .)

The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. Even more generally, one can study sequences with elements in some topological space.

[edit] Series

The sum of a sequence is a series. More precisely, if $(x 1, x 2, x 3,...)$ is a sequence, one may consider the sequence of partial sums $(S 1, S 2, S 3,...)$ , with

$S_n=x_1+x_2+\dots + x_n=\sum\limits_{i=1}^{n}x_i.$

Formally, this pair of sequences comprises the series with the terms $x 1, x 2, x 3,...$ , which is denoted as

$\sum\limits_{i=1}^{\infty}x_i.$

If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see series.

[edit] Infinite sequences in theoretical computer science

Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as sequences (as opposed to finite strings). Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0,1}). The set $\mathcal{C}=\{0,1\}^\infty$ of all infinite, binary sequences is sometimes called the Cantor space.

An infinite binary sequence can represent a formal language (a set of strings) by setting the nth bit of the sequence to 1 if and only if the nth string (in shortlex order) is in the language. Therefore, the study of complexity classes, which are sets of languages, may be regarded as studying sets of infinite sequences.

An infinite sequence drawn from the alphabet {0,1,...,b-1} may also represent a real number expressed in the base-b positional number system. This equivalence is often used to bring the techniques of real analysis to bear on complexity classes.

[edit] Sequences as vectors

Sequences over a field may also be viewed as vectors in a vector space. Specifically, the set of F-valued sequences (where F is a field) is a function space (in fact, a product space) of F-valued functions over the set of natural numbers.

[edit] Doubly-infinite sequences

Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other -- the sequence has a first element, but no final element (a singly-infinite sequence). A doubly-infinite sequence is infinite in both directions -- it has no final element, but it has no first element either. Singly-infinite sequences are functions from the natural numbers (N) to some set, whereas doubly-infinite sequences are functions from the integers (Z) to some set.