Information AboutSequence |
| CATEGORIES ABOUT SEQUENCE | |
| elementary mathematics | |
| sequences | |
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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 the exact 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 Integer s (2, 4, 6,...). 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 (''a''1, ''a''2, ...). 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 Number s without 0) to ''S''. Sequences may also start from 0, so the first term in the sequence is then ''a''0. 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. 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 ''non-decreasing'' and ''non-increasing'' are used in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively. If the terms of a sequence are Integer s, then the sequence is an Integer Sequence . If the terms of a sequence are Polynomial s, 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 . 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 Number s. (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 Number s. This type can be generalized to sequences of elements of some Vector Space . In analysis, the vector spaces considered are often Function Space s. Even more generally, one can study sequences with elements in some Topological Space . SERIES See Also: Series (mathematics) The sum of a sequence is a Series . More precisely, if (''x''1, ''x''2, ''x''3, ...) is a sequence, one may consider the sequence of Partial Sum s (''S''1, ''S''2, ''S''3, ...), with : Formally, this pair of sequences comprises the ''series'' with the terms ''x''1, ''x''2, ''x''3, ..., which is denoted as : If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see Series . 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 . They are often referred to simply as ''sequences'' (as opposed to finite '' Strings ''). Infinite binary sequences, for instance, are infinite sequences of Bit s (characters drawn from the alphabet {0,1}). The set ''C'' = {0, 1}∞ of all infinite, binary sequences is sometimes called the Cantor Space . An infinite binary sequence can represent a Formal Language (a set of strings) by setting the ''n'' th bit of the sequence to 1 if and only if the ''n'' th string (in Shortlex Order ) is in the language. Therefore, the study of Complexity Class es, which are sets of languages, may be regarded as studying sets of infinite sequences. 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 . This equivalence is often used to bring the techniques of Real Analysis to bear on complexity classes. SEQUENCES AS VECTORS Sequences over a field may also be viewed as Vectors in a Vector Space . Specifically, the set of ''F''-valued sequences (where ''F'' is a Field ) is a Function Space (in fact, a Product Space ) of ''F''-valued functions over the set of natural numbers. In particular, the term '' Sequence Space '' usually refers to a Linear Subspace of the set of all possible infinite sequences with elements in . DOUBLY-INFINITE SEQUENCES Normally, the term ''infinite sequence'' refers to a sequence which is infinite in one direction, and finite in the other -- the sequence has a first element, but no final element (a ''singly-infinite sequence''). A ''doubly-infinite sequence'' is infinite in both directions -- it has no final element, but it has no first element either. Singly-infinite sequences are functions from the natural numbers (N') to some set, whereas doubly-infinite sequences are functions from the integers ('''Z''') to some set. One can interpret singly infinite sequences as element of the Semigroup Ring of the Natural Numbers |
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