sequence
Sign in to savethumb|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
AI overview
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.
AI-generated from the Wikipedia summary — may contain errors.
Article
34 sectionsContents
- 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
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, 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 , 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 .