Modulus of convergence

In the study of real analysis, 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'.

References

• Klaus Weihrauch (2000), Computable Analysis.