Modulus of convergence explained

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 .

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 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

• Modulus of continuity

References

• Klaus Weihrauch (2000), Computable Analysis.