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

Definition

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

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

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

See also

• Modulus of continuity

References

• Klaus Weihrauch (2000), Computable Analysis.