If a sequence of real numbers converges to a real number , then by definition, for every real there is a natural number such that if then . A modulus of convergence is essentially a function that, given , returns a corresponding value of .
DefinitionEdit
Suppose that is a convergent sequence of real numbers with limit. There are two ways of defining a modulus of convergence as a function from natural numbers to natural numbers:
As a function such that for all , if then .
As a function such that for all , if then .
The latter definition is often employed in constructive settings, where the limit may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces with .
modulus, convergence, real, analysis, branch, mathematics, modulus, convergence, function, that, tells, quickly, convergent, sequence, converges, these, moduli, often, employed, study, computable, analysis, constructive, mathematics, sequence, real, numbers, d. 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 displaystyle x i converges to a real number x displaystyle x then by definition for every real e gt 0 displaystyle varepsilon gt 0 there is a natural number N displaystyle N such that if i gt N displaystyle i gt N then x x i lt e displaystyle left x x i right lt varepsilon A modulus of convergence is essentially a function that given e displaystyle varepsilon returns a corresponding value of N displaystyle N Definition EditSuppose that x i displaystyle x i is a convergent sequence of real numbers with limit x displaystyle x There are two ways of defining a modulus of convergence as a function from natural numbers to natural numbers As a function f displaystyle f such that for all n displaystyle n if i gt f n displaystyle i gt f n then x x i lt 1 n displaystyle left x x i right lt 1 n As a function g displaystyle g such that for all n displaystyle n if i j gt g n displaystyle i geq j gt g n then x i x j lt 1 n displaystyle left x i x j right lt 1 n The latter definition is often employed in constructive settings where the limit x displaystyle x may actually be identified with the convergent sequence Some authors use an alternate definition that replaces 1 n displaystyle 1 n with 2 n displaystyle 2 n See also EditModulus of continuityReferences EditKlaus Weihrauch 2000 Computable Analysis Retrieved from https en wikipedia org w index php title Modulus of convergence amp oldid 1089198925, wikipedia, wiki, book, books, library,