Question

Cauchy sequences?

Answer 1

For the most part, it's just a fancy name for convergent sequences :D

Answer 2

so what's the question @SOMU.kvp

Answer 3

need easy definition for this?

Answer 4

Easy mathematical or easy definition that someone not familiar with maths terms would get? :D

Answer 5

Formal mathematical definition would be "A sequence \(\large x_n\) is a Cauchy sequence if for any \(\large\varepsilon > 0\) there exists a positive integer \(\large N\) such that for all integers \(\large m,n\) where \(\large m > n\) and \(\large n>N\) then
\[\Large |x_m-x_n|<\varepsilon\]

Answer 6

A rough translation into "English" would probably have it... a sequence of numbers is a Cauchy sequence if for any small number, \(\large \varepsilon\), at some point in the sequence, the terms are no more than \(\large \varepsilon\) units apart :)

Answer 7

@terenzreignz What is the "N" there for? It seems to serve no purpose in the definition.

Answer 8

actually, I have a typo :3
should be \[\large m>N\]
Sorry :)

The N there corresponds to the "at some point" in the "English" definition :)

Answer 9

Means they don't have to be no more than epsilon units apart from the beginning of the sequence... just at some point, and onward :D

Answer 10

It's similar to the big N in the definition of a convergent sequence :)

A sequence \(\large x_n\) is convergent to \(\large x\) if for any \(\large \varepsilon >0\) there exists a positive integer \(\large N\) such that for all integers \(\large n>N\) then
\[\Large |x_n - x|< \varepsilon\]

Answer 11

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(a) The plot of a Cauchy sequence (x_n), shown in blue, as x_n versus n If the space containing the sequence is complete, the "ultimate destination" of this sequence (that is, the limit) exists.
(b) A sequence that is not Cauchy. The elements of the sequence fail to get arbitrarily close to each other as the sequence progresses.

In mathematics, a Cauchy sequence (pronounced [koʃiˈ]), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.[1] More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

The utility of Cauchy sequences lies in the fact that in a complete metric space (one where all such sequences are known to converge to a limit), the criterion for convergence depends only on the terms of the sequence itself. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates, thus fulfilling a logical condition, such as termination.

The notions above are not as unfamiliar as they might at first appear. The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers (whose terms are the successive truncations of the decimal expansion of x) has the real limit x. In some cases it may be difficult to describe x independently of such a limiting process involving rational numbers.

Answer 12

http://www.proofwiki.org/wiki/Definition:Cauchy_Sequence