Modulus of convergence

In real analysis, a convergence module is a function that indicates how fast a convergent sequence converges. Convergence modules are often used in computable analysis and constructive mathematics.

If a sequence of real numbers converges to a real number, then there is, by definition, for every real a natural number so that if. A convergence module is essentially a function that, given a corresponding calculated value.

Definition

Be a convergent sequence of real numbers with limit. There are two ways to define a convergence module as a function of the natural numbers to the natural numbers:

• As a function such that for all cases: if, then.
• As a function such that for all cases: if, then. (This exists since every convergent sequence is a Cauchy sequence. )

The latter definition is often used in design scenarios, wherein the threshold is identified in certain circumstances with the convergent sequence. Some authors use an alternative definition which replaced against.