Sequence

In mathematics, 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. A sequence is a discrete function.

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,...). Finite sequences are sometimes known as strings or words, and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

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.

In addition to identifying the elements of a sequence by their position, such as "the 3rd element", elements may be given names for convenient referencing. For example a sequence might be written as (a1, a2, a2, … ), or (b0, b1, b2, … ), or (c0, c2, c4, … ), depending on what is useful in the application.

Finite and infinite

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, ... } to S. For example, the sequence of prime numbers (2,3,5,7,11, … ) is the function 1→2, 2→3, 3→5, 4→7, 5→11, … .

A sequence of a finite length n 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 or two-way infinite sequence. An example is the bi-infinite sequence of all even integers ( … , -4, -2, 0, 2, 4, … ).

Multiplicative

Let A=(a1, a2, a3, ...) be a sequence defined by a function f:{1, 2, 3, ...} → {1, 2, 3, ...}, such that a i = f(i).

The sequence is multiplicative if f(xy) = f(x)f(y) for all x,y such that x and y are coprime.

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 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 monotonic function.

The terms nondecreasing and nonincreasing are 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, then it becomes possible to consider convergence of an infinite sequence in S. Such considerations involve the concept of the limit of a sequence.

If A is a set, the free monoid over A (denoted A*) is a monoid containing all the finite sequences (or strings) of zero or more elements drawn from A, with the binary operation of concatenation. The free semigroup A+ is the subsemigroup of A* containing all elements except the empty sequence.

Sequences in analysis

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


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 xn = 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. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. Even more generally, one can study sequences with elements in some topological space.

Series

The sum of terms of a sequence is a series. More precisely, if (x1, x2, x3, ...) is a sequence, one may consider the sequence of partial sums (S1, S2, S3, ...), with]

Formally, this pair of sequences comprises the series with the terms x1, x2, x3, ..., which is denoted as




If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit.

Reference:
http://en.wikipedia.org/wiki/Sequence