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 (xi) 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 − xi| < ε. A modulus of convergence is essentially a function that, given ε, returns a corresponding value of N.
Suppose that (xi) 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 − xi| < 1/n
- As a function g(n) such that for all n, if i ≥ j > g(n) then |xi − xj| < 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.
- Klaus Weihrauch (2000), Computable Analysis.