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 converges to a real number, then by definition, for every real there is a natural number such that if then. A modulus of convergence is essentially a function that, given, returns a corresponding value of.

Examples

Suppose that is a convergent sequence of real numbers with limit. There are two common ways of defining a modulus of convergence as a function from natural numbers to natural numbers:

As a function such that for all, if then.
As a function such that for all, if then.

The latter definition is often employed in constructive settings, where the limit may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces with.