From c.850 to c.1000, a non-biblical Latin text added to a long portion of chant originally sung to one syllable at the end of the Alleluia; later, a similar syllabic chant specially composed. All but four sequences (they include the Dies irae of the Requiem Mass) were banned from the liturgy in the 16th-c, but the Stabat mater was later admitted.
For example, (C,R,Y) is a sequence of letters that differs from (Y,C,R), as the ordering matters.
The members of a sequence are also called its elements or terms, and the number of terms (possibly infinite) is called the length of the sequence.
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 finite sequence is also called an n-tuple. 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.
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 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 is possible to talk about convergence of an infinite sequence in S.
Sequences in analysis
In mathematical analysis, when talking about sequences, one usually understands sequences of the form
(x1,x2,x3,...) or (x0,x1,x2,...)i.e. 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.
Series
The sum 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.
Infinite sequences in theoretical computer science
Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science.
An infinite binary sequence can represent a formal language (a set of strings) by setting the nth bit of the sequence to 1 if and only if the nth string (in lexicographical order) is in the language.
An infinite sequence drawn from the alphabet {0,1,...,b-1} may also represent a real number expressed in the base-b positional number system.
Sequences as vectors
Sequences over a field may also be viewed as vectors in a vector space.
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