Convergents of a continued fraction

Definition

Let $[a_0; a_1, a_2, \dots]$ be a simple continued fraction. For each $n$ such that $a_n$ exists, the $n$-th convergent is the rational number $\dfrac{p_n}{q_n} = [a_0; a_1, \dots, a_n]$. Here $p_n$ and $q_n$ are given by $p_{-2} = 0$, $p_{-1} = 1$, $q_{-2} = 1$, $q_{-1} = 0$ and $$p_n = a_n p_{n-1} + p_{n-2}, \qquad q_n = a_n q_{n-1} + q_{n-2} \quad (n \ge 0).$$

Remarks

The recurrences give $p_n q_{n-1} - p_{n-1} q_n = (-1)^{n-1}$. So $\gcd(p_n, q_n) = 1$, and the fraction $p_n / q_n$ is in lowest terms. For $\sqrt{2} = [1; 2, 2, \dots]$, the convergents are $1, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29}, \dots$. For a general continued fraction, the same recurrences hold with $p_n = b_n p_{n-1} + a_n p_{n-2}$ and $q_n = b_n q_{n-1} + a_n q_{n-2}$.

Explanations

L1 human

The convergents are the fractions that you get when you cut a continued fraction off after finitely many terms. They approach the value of the continued fraction, alternately from below and from above.