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 x i {\displaystyle x_{i}} converges to a real number x {\displaystyle x} , then by definition, for every real ε > 0 {\displaystyle \varepsilon >0} there is a natural number N {\displaystyle N} such that if i > N {\displaystyle i>N} then | x − x i | < ε {\displaystyle \left|x-x_{i}\right|<\varepsilon } . A modulus of convergence is essentially a function that, given ε {\displaystyle \varepsilon } , returns a corresponding value of N {\displaystyle N} .

Suppose that x i {\displaystyle x_{i}} is a convergent sequence of real numbers with limit x {\displaystyle x} . There are two common ways of defining a modulus of convergence as a function from natural numbers to natural numbers:

• As a function f {\displaystyle f} such that for all n {\displaystyle n} , if i > f ( n ) {\displaystyle i>f(n)} then | x − x i | < 1 / n {\displaystyle \left|x-x_{i}\right|<1/n} .
• As a function g {\displaystyle g} such that for all n {\displaystyle n} , if i ≥ j > g ( n ) {\displaystyle i\geq j>g(n)} then | x i − x j | < 1 / n {\displaystyle \left|x_{i}-x_{j}\right|<1/n} .

The latter definition is often employed in constructive settings, where the limit x {\displaystyle x} may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces 1 / n {\displaystyle 1/n} with 2 − n {\displaystyle 2^{-n}} .

• Modulus of continuity

• Klaus Weihrauch (2000), Computable Analysis.