Modulus of convergence
Encyclopedia
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.
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.
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
, 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
Computable analysis
In mathematics, computable analysis is the study of which parts of real analysis and functional analysis can be carried out in a computable manner. It is closely related to constructive analysis.- Basic results :...
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.
Definition
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.