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

Suppose that ${\displaystyle x_{i}}${\displaystyle x_{i}} is a convergent sequence of real numbers with limit ${\displaystyle x}${\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 ${\displaystyle f}${\displaystyle f} such that for all ${\displaystyle n}${\displaystyle n}, if ${\displaystyle i>f(n)}${\displaystyle i>f(n)} then ${\displaystyle \left|x-x_{i}\right|<1/n}${\displaystyle \left|x-x_{i}\right|<1/n}.
• As a function ${\displaystyle g}${\displaystyle g} such that for all ${\displaystyle n}${\displaystyle n}, if ${\displaystyle i\geq j>g(n)}${\displaystyle i\geq j>g(n)} then ${\displaystyle \left|x_{i}-x_{j}\right|<1/n}${\displaystyle \left|x_{i}-x_{j}\right|<1/n}.

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

• Modulus of continuity

• Klaus Weihrauch (2000), Computable Analysis.

• Constructivism (mathematics)
• Computable analysis
• Real analysis

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