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

• Modulus of continuity

References

• Klaus Weihrauch (2000), Computable Analysis.