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 \varepsilon > 0 there is a natural number N such that if i > N then \left|x - x_i\right| < \varepsilon. A modulus of convergence is essentially a function that, given \varepsilon, returns a corresponding value of N.

Examples

Suppose that x_i is a convergent sequence of real numbers with limit x. There are two common ways of defining a modulus of convergence as a function from natural numbers to natural numbers:

  • As a function f such that for all n, if i > f(n) then \left|x - x_i\right| < 1/n.
  • As a function g such that for all n, if i \geq j > g(n) then \left|x_i - x_j\right| < 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.