sequence

Also known as ordered list, ordered set

thumb|A part of an infinite sequence of real numbers (in blue), indexed by a natural number . This sequence is neither increasing, decreasing, convergent, nor Cauchy. It is, however, bounded (by red dashed lines). In mathematics, a sequence is a collection of objects possibly with repetition, that come in a specified order. Like a set, it contains members (also called elements, or terms). Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. The notion of a sequence can be generalized to an indexed family, defin

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A sequence is an ordered collection of objects where the same element can appear multiple times and the position of each element matters, distinguishing it from a simple set. Sequences are important in mathematics because they allow us to study patterns, repetition, and ordered relationships—such as whether a sequence grows, shrinks, converges to a value, or stays within certain bounds.

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Commons gallery: Sequence
Commons category: Sequence
equivalent class: schema.org/Series
P8168: Ali Akbar Velayati

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~28 min read

Article

34 sections

Contents

Examples and notation
Examples
Indexing
Defining a sequence by recursion
Formal definition and basic properties
Definition
Finite and infinite
Increasing and decreasing
Bounded
Subsequences
Other types of sequences
Limits and convergence
Formal definition of convergence
Applications and important results
Cauchy sequences
Infinite limits
Series
Use in other fields of mathematics
Topology
Product topology
Analysis
Sequence spaces
Linear algebra
Abstract algebra
Free monoid
Exact sequences
Spectral sequences
Set theory
Computing
Streams
See also
Notes
References
External links

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 , which contains the number at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of positive even integers .