Convergents converge to the number

Builds on Convergents of a continued fraction Continued fraction expansion of a real number

Statement

Let $x$ be an irrational real number, and let $p_n / q_n$ be the convergents of its continued fraction expansion. Then $\left| x - \frac{p_n}{q_n} \right| < \frac{1}{q_n q_{n+1}} \le \frac{1}{q_n^2}$ for each $n \ge 0$ , and $p_n / q_n \to x$ as $n \to \infty$ .

Remarks

The convergents with even $n$ increase toward $x$ , and the convergents with odd $n$ decrease toward $x$ . For a rational $x$ , the last convergent is equal to $x$ . Conversely, each infinite simple continued fraction converges to an irrational number, and this number has that continued fraction as its expansion. For example, $\frac{355}{113}$ is a convergent of $\pi$ , and $\left| \pi - \frac{355}{113} \right| < 3 \cdot 10^{-7}$ .

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Continued fraction definition
Simple continued fraction definition
Continued fraction expansion of a real number definition
Convergents of a continued fraction definition
Convergents converge to the number statement

Tag	`0E70M92FE3`
URI	`m2://0E70M92FE3`
Kind	statement, theorem
Writing	W0 stub
MSC	11A55
Status	This copy of the site has no build data. Run `m2 status 0E70M92FE3`.
Uses	Convergents of a continued fraction, Continued fraction expansion of a real number

Convergents converge to the number

Statement

Remarks

Reading path

Further reading

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