# 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 ε > 0 there is a natural number *N* such that if *i* > *N* then |*x* − *x*_{i}| < ε. A modulus of convergence is essentially a function that, given ε, 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*(*n*) such that for all*n*, if*i*>*f*(*n*) then |*x*−*x*_{i}| < 1/*n* - As a function
*g*(*n*) such that for all*n*, if*i*≥*j*>*g*(*n*) then |*x*_{i}−*x*_{j}| < 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

## References

- Klaus Weihrauch (2000),
*Computable Analysis*.