Accuracy of Convergents of Continued Fraction

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Accuracy of Convergents of Continued Fraction Expansion of Irrational Number

Let $x$ be an irrational number.

Let $(a_0, a_1, \ldots)$ be its continued fraction expansion.

Let $\left \langle {C_n}\right \rangle_{n \geq 0}$ be its sequence of convergents.

Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators.


$\forall k \ge 1: \left\vert{x - \dfrac {p_{k + 1} } {q_{k + 1} } }\right\vert < \dfrac 1 {q_{k + 1} q_{k + 2} } \le \dfrac 1 {2 q_k q_{k + 1} } < \left\vert{x - \dfrac {p_k} {q_k} }\right\vert$


The left hand side of the inequality gives an indication of how close each convergent gets to its true value.
The right hand side gives a bound that limits its accuracy.

Accuracy of Convergents of Convergent Simple Infinite Continued Fraction

Let $C = \tuple {a_0, a_1, \ldots}$ be an simple infinite continued fraction in $\R$.

Let $C$ converge to $x \in \R$.

For $n \ge 0$, let $C_n = \dfrac {p_n} {q_n}$ be the $n$th convergent of $C$, where $p_n$ and $q_n$ are the $n$th numerator and denominator.

Then for all $n \ge 0$:

$\size {x - \dfrac {p_n} {q_n} } < \dfrac 1 {q_n q_{n + 1} }$


We have a posteriori, by Correspondence between Irrational Numbers and Simple Infinite Continued Fractions, that the statements above coincide, or at least partially.

The point is that they are used when proving the correspondence.

Also see