Modulus of convergence
In real analysis, a branch of mathematics, a modulus of convergence is a 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 numbers [math]\displaystyle{ x_i }[/math] converges to a real number [math]\displaystyle{ x }[/math], then by definition, for every real [math]\displaystyle{ \varepsilon \gt 0 }[/math] there is a natural number [math]\displaystyle{ N }[/math] such that if [math]\displaystyle{ i \gt N }[/math] then [math]\displaystyle{ \left|x - x_i\right| \lt \varepsilon }[/math]. A modulus of convergence is essentially a function that, given [math]\displaystyle{ \varepsilon }[/math], returns a corresponding value of [math]\displaystyle{ N }[/math].
Definition
Suppose that [math]\displaystyle{ x_i }[/math] is a convergent sequence of real numbers with limit [math]\displaystyle{ 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]\displaystyle{ f }[/math] such that for all [math]\displaystyle{ n }[/math], if [math]\displaystyle{ i \gt f(n) }[/math] then [math]\displaystyle{ \left|x - x_i\right| \lt 1/n }[/math].
- As a function [math]\displaystyle{ g }[/math] such that for all [math]\displaystyle{ n }[/math], if [math]\displaystyle{ i \geq j \gt g(n) }[/math] then [math]\displaystyle{ \left|x_i - x_j\right| \lt 1/n }[/math].
The latter definition is often employed in constructive settings, where the limit [math]\displaystyle{ x }[/math] may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces [math]\displaystyle{ 1/n }[/math] with [math]\displaystyle{ 2^{-n} }[/math].
See also
References
- Klaus Weihrauch (2000), Computable Analysis.
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