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This is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical).

In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, or after, every other member. A sequence is a function with a domain equal to the set of positive integers.

The sequence of positive integers is: 1, 2, 3, ..., n - 1, n, n + 1, ... Each number is a term, with n being the "n-th term". A sequence can be denoted by: {an}; such that, in the above list of positive integers, a1 is 1, a317 is 317, and an is n. This is also indicated by: a0, a1, a2, ..., an, ... The terms of a sequence, are part of a set, commonly indicated by S; they are a sequence in S.

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 numbersNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("this) into some set S.

If S is the set of integers, then the sequence is an integer sequenceNumber theory In mathematics, an integer sequence is a sequence (i. an ordered list) of integers. An integer sequence may be specified explicitly by giving a formula for its n th term, or implicitly by giving a relationship between its terms. For example,. If S is a set of polynomialIn mathematics polynomial functions or polynomials are an important class of simple and smooth functions. Simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i. they have derivatives os, the sequence is a polynomial sequenceIn mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3,. in which each index is equal to the degree of the corresponding polynomial. Various special polynomial sequences are known by eponyms; amon.

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.

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

The sum of a sequence is a series. For example:



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Topics: Sequence, Infinitesimal Calculus, Infinite Descending Chain...

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