Modulus of convergence

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: 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 $$2^{-n}$$.
 * 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 $$\left|x_i - x_j\right| < 1/n$$.