Inreal analysis, a branch ofmathematics, amodulus of convergence is afunction that tells how quickly aconvergent sequence converges. These moduli are often employed in the study ofcomputable analysis andconstructive mathematics.

If a sequence ofreal numbers${\displaystyle x_i}$ converges to a real number${\displaystyle x}$, then by definition, for every real${\displaystyle \epsilon >0}$ there is anatural number${\displaystyle N}$ such that if${\displaystyle i>N}$ then. A modulus of convergence is essentially a function that, given${\displaystyle \epsilon }$, returns a corresponding value of${\displaystyle N}$.

Suppose that${\displaystyle x_i}$ is a convergent sequence of real numbers withlimit${\displaystyle x}$. There are two common ways of defining a modulus of convergence as a function from natural numbers to natural numbers:

• As a function${\displaystyle f}$ such that for all${\displaystyle n}$, if${\displaystyle i>f(n)}$ then.
• As a function${\displaystyle g}$ such that for all${\displaystyle n}$, if${\displaystyle i\ge j>g(n)}$ then.

The latter definition is often employed in constructive settings, where the limit${\displaystyle x}$ may actually be identified with the convergent sequence. Some authors use an alternate definition that replaces${\displaystyle 1/n}$ with${\displaystyle 2^{-n}}$.

• Modulus of continuity

• Klaus Weihrauch (2000),Computable Analysis.

• Constructivism (philosophy of mathematics)
• Computable analysis
• Real analysis

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