Sequence

There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. One way to specify a sequence is to list all its elements. For example, the first four odd numbers form the sequence (1, 3, 5, 7). This notation is used for infinite sequences as well. For instance, the infinite sequence of positive odd integers is written as (1, 3, 5, 7, ...). Because notating sequences with ellipsis leads to ambiguity, listing is most useful for customary infinite sequences which can be easily recognized from their first few elements. Other ways of denoting a sequence are discussed after the examples.

In some cases, the elements of the sequence are related naturally to a sequence of integers whose pattern can be easily inferred. In these cases, the index set may be implied by a listing of the first few abstract elements. For instance, the sequence of squares of odd numbers could be denoted in any of the following ways.

Sequences whose elements are related to the previous elements in a straightforward way are often defined using recursion. This is in contrast to the definition of sequences of elements as functions of their positions.

To define a sequence by recursion, one needs a rule, called recurrence relation to construct each element in terms of the ones before it. In addition, enough initial elements must be provided so that all subsequent elements of the sequence can be computed by successive applications of the recurrence relation.

The Fibonacci sequence is a simple classical example, defined by the recurrence relation

A linear recurrence with constant coefficients is a recurrence relation of the form

A holonomic sequence is a sequence defined by a recurrence relation of the form

Not all sequences can be specified by a recurrence relation. An example is the sequence of prime numbers in their natural order (2, 3, 5, 7, 11, 13, 17, ...).

The length of a sequence is defined as the number of terms in the sequence.

A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy.

Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called complete metric spaces and are particularly nice for analysis.

Sequences play an important role in topology, especially in the study of metric spaces. For instance:

In analysis, when talking about sequences, one will generally consider sequences of the form

which is to say, infinite sequences of elements indexed by natural numbers.

Abstract algebra employs several types of sequences, including sequences of mathematical objects such as groups or rings.

The sequence of groups and homomorphisms may be either finite or infinite.