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

Definition

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

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

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

See also

• Modulus of continuity

References

• Klaus Weihrauch (2000), Computable Analysis.