Search results
Results from the WOW.Com Content Network
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 ...
Cauchy's convergence test can only be used in complete metric spaces (such as and ), which are spaces where all Cauchy sequences converge. This is because we need only show that its elements become arbitrarily close to each other after a finite progression in the sequence to prove the series converges.
By construction, every real number x is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges to x is a Cauchy sequence representing x. This reflects the observation that one can often use different sequences to approximate the same real number. [6]
The Cauchy distribution is an example of a distribution which has no mean, variance or higher moments defined. Its mode and median are well defined and are both equal to . The Cauchy distribution is an infinitely divisible probability distribution. It is also a strictly stable distribution. [8]
2 Examples. 3 Convergence of products. 4 See also. ... This is also known as the nth root test or Cauchy's criterion. ... each element of the two sequences is ...
Here the nth term in the sequence is the nth decimal approximation for pi. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number. (In this real number line, this sequence converges to pi.) Cauchy completeness is related to the construction of the real numbers using Cauchy sequences.
It can be shown that a real-valued sequence is Cauchy if and only if it is convergent. This property of the real numbers is expressed by saying that the real numbers endowed with the standard metric, (, | |), is a complete metric space. In a general metric space, however, a Cauchy sequence need not converge.
The Cauchy product may apply to infinite series [1] [2] or power series. [3] [4] When people apply it to finite sequences [5] or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section.