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Sequence

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 arranged in a linear fashion; such that each member comes either before, or after, every other member, and the order of members is important.

For example, (C,Y,R) is a sequence of letters; the ordering is that C is first, Y is second, and R is third. Sequences can be finite, as in the example just given, or infinite, such as the sequence of all even positive integers (2,4,6,...). Finite sequences include the null sequence ( ) that has no elements. The elements in a sequence are also called terms, and the number of terms (possibly infinite) is called the length of the sequence.

A sequence is denoted (a1,a2, ...). For shortness, the notation (an) 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.

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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 a 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.

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.

Series

The sum of a sequence of real numbers is a series. Alternately stated, a series is a sequence of partial sums. For example:

$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots+\frac{1}{2^{n-1}} = \frac{2^n-1}{2^{n-1}}.$