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Some other types of sequences that are easy to define include: An integer sequence is a sequence whose terms are integers. A polynomial sequence is a sequence whose terms are polynomials. A positive integer sequence is sometimes called multiplicative, if a nm = a n a m for all pairs n, m such that n and m are coprime. [8]
Since the topological vector space definition of Cauchy sequence requires only that there be a continuous "subtraction" operation, it can just as well be stated in the context of a topological group: A sequence () in a topological group is a Cauchy sequence if for every open neighbourhood of the identity in there exists some number such that ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients. Some specific examples that are close, in some sense, to the Fibonacci sequence include:
For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2. Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is
A sequence that does not converge is said to be divergent. [3] The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. [1] Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.
Oscillation of a sequence (shown in blue) is the difference between the limit superior and limit inferior of the sequence. In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point.
An integer sequence is a definable sequence relative to M if there exists some formula P(x) in the language of set theory, with one free variable and no parameters, which is true in M for that integer sequence and false in M for all other integer sequences.