Accuracy of Convergents of Continued Fraction
Theorem
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.
Then:
- $\forall k \ge 1: \left\vert{x - \dfrac {p_{k + 1} } {q_{k + 1} } }\right\vert < \dfrac 1 {q_{k + 1} q_{k + 2} } \le \dfrac 1 {2 q_k q_{k + 1} } < \left\vert{x - \dfrac {p_k} {q_k} }\right\vert$
Thus:
- 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.
Convergent simple infinite continued fraction
Let $C = \tuple {a_0, a_1, \ldots}$ be an infinite simple 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 $n \ge 0$:
- $\size {x - \dfrac {p_n} {q_n} } < \dfrac 1 {q_n q_{n + 1} }$.
Caution
A posteriori, by Correspondence between Irrational Numbers and Simple Infinite Continued Fractions, the statements above coincide, or at least partially. The point is that they are used when proving the correspondence.