Calculus/Sequences

A sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence.

For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...).

An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent. It is however bounded.

Examples and notation

There are various and quite different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the notations introduced below.

A sequence may be denoted (a₁, a₂, ...). For shortness, the notation (a_n) 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 without 0) to S.

Sequences may also start from 0, so the first term in the sequence is then a₀.

A finite sequence is also called an n-tuple. Finite sequences include the empty sequence ( ) that has no elements.

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.

The sequence $a_{n}\ =\ {\frac {1}{n}}$ is called the harmonic sequence.

If c and d are given numbers, the sequence $a_{n}\ =\ c+nd$  is an arithmetic sequence.

If b and r ≠ 0 are given, the sequence $a_{n}\ =\ br^{n}$  is a geometric sequence.

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 (which, as stated in the introduction, can also be called "terms") without disturbing the relative positions of the remaining elements.

If the terms of the sequence are a subset of an 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 a monotonic function. A sequence that both increases and decreases (at different places in the sequence) is said to be non-monotonic or non-monotone.

The terms non-decreasing and non-increasing are often used in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively. 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 (as is true of real numbers, for example), then it becomes possible to consider the convergence of an infinite sequence in S. Such considerations involve the concept of the limit of a sequence.

It can be shown that bounded monotonic sequences must converge.

Sequences in analysis

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

$(x_{1},x_{2},x_{3},...)\,$ or $(x_{0},x_{1},x_{2},...)\,$

which is to say, infinite sequences of elements indexed by natural numbers. (It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by x_n = 1/log(n) would be defined only for n ≥ 2. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given N.)

The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers.