View source for Modulus of convergence

{{Short description|Mathematical term}}
In [[real analysis]], a branch of [[mathematics]], a '''modulus of convergence''' is a [[Function (mathematics)|function]] that tells how quickly a [[convergent sequence]] converges.  These moduli are often employed in the study of [[computable analysis]] and [[constructive mathematics]]. 

If a sequence of [[real number]]s <math>x_i</math> converges to a real number <math>x</math>, then by definition, for every real <math>\varepsilon > 0</math> there is a [[natural number]] <math>N</math> such that if <math>i > N</math> then <math>\left|x - x_i\right| < \varepsilon</math>. A modulus of convergence is essentially a function that, given <math>\varepsilon</math>, returns a corresponding value of <math>N</math>.

== Definition ==

Suppose that <math>x_i</math> is a convergent sequence of real numbers with [[Limit of a sequence|limit]] <math>x</math>. There are two ways of defining a modulus of convergence as a function from natural numbers to natural numbers:
* As a function <math>f</math> such that for all <math>n</math>, if <math>i > f(n)</math> then <math>\left|x - x_i\right| < 1/n</math>.
* As a function <math>g</math> such that for all <math>n</math>, if <math>i \geq j > g(n)</math> then <math>\left|x_i - x_j\right| < 1/n</math>.
The latter definition is often employed in constructive settings, where the limit <math>x</math> may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces <math>1/n</math> with <math>2^{-n}</math>.

== See also ==
* [[Modulus of continuity]]

== References ==
* Klaus Weihrauch (2000), ''Computable Analysis''.

[[Category:Constructivism (mathematics)]]
[[Category:Computable analysis]]
[[Category:Real analysis]]