Convergence module

In real analysis , a convergence module is a function that indicates how quickly a convergent sequence converges. Convergence modules are often used in computational analysis and constructive mathematics .

If a sequence of real numbers converges to a real number , then by definition there is a natural number for every real number such that , if . A convergence module is essentially a function that, given a given, calculates a corresponding value of . ${\displaystyle (x_{i})}$ ${\displaystyle x}$ ${\displaystyle \epsilon >0}$ ${\displaystyle N}$ ${\displaystyle \left|x-x_{i}\right|<\epsilon }$ ${\displaystyle i>N}$ ${\displaystyle \epsilon }$ ${\displaystyle N}$

definition

Let be a convergent sequence of real numbers with limit . There are two ways to define a module of convergence as a function from the natural numbers to the natural numbers: ${\displaystyle (x_{i})}$ ${\displaystyle x}$

• As a function in such a way that applies to everyone : if , then . ${\displaystyle f(n)}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle i>f(n)}$ ${\displaystyle \left|x-x_{i}\right|<1/n}$
• As a function in such a way that applies to everyone : if , then . (This exists because every convergent sequence is a Cauchy sequence .) ${\displaystyle g(n)}$ ${\displaystyle n}$ ${\displaystyle i\geq j>g(n)}$ ${\displaystyle \left|x_{i}-x_{j}\right|<1/n}$

The latter definition is often used in constructive scenarios, where the limit value may be identified with the convergent sequence. Some authors use an alternative definition that is replaced by. ${\displaystyle x}$ ${\displaystyle 1/n}$ ${\displaystyle 2^{-n}}$

Individual evidence

• Klaus Weihrauch (2000), Computable Analysis .