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  "body": "\n## Definition\n\nLet $[a_0; a_1, a_2, \\dots]$ be a [[simple continued fraction|06XF7BCFFJ]].\nFor each $n$ such that $a_n$ exists, the $n$-th convergent is the rational number $\\dfrac{p_n}{q_n} = [a_0; a_1, \\dots, a_n]$.\nHere $p_n$ and $q_n$ are given by $p_{-2} = 0$, $p_{-1} = 1$, $q_{-2} = 1$, $q_{-1} = 0$ and\n$$p_n = a_n p_{n-1} + p_{n-2}, \\qquad q_n = a_n q_{n-1} + q_{n-2} \\quad (n \\ge 0).$$\n\n## Remarks\n\nThe 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.\nFor $\\sqrt{2} = [1; 2, 2, \\dots]$, the convergents are $1, \\frac{3}{2}, \\frac{7}{5}, \\frac{17}{12}, \\frac{41}{29}, \\dots$.\nFor a general [[continued fraction|0BK6JBK58W]], the same recurrences hold with $p_n = b_n p_{n-1} + a_n p_{n-2}$ and $q_n = b_n q_{n-1} + a_n q_{n-2}$.\n\n## Explanations\n\n### L1 human\n\nThe convergents are the fractions that you get when you cut a continued fraction off after finitely many terms.\nThey approach the value of the continued fraction, alternately from below and from above.\n",
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