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  "body": "\n## Statement\n\nLet $b_0, b_1, b_2, \\dots$ be complex numbers and let $a_1, a_2, \\dots$ be nonzero complex numbers.\nConsider the [[continued fraction|0BK6JBK58W]] $b_0 + \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + \\cdots}}$, and let $x_n$ be its $n$-th [[convergent|0SRWNR42BQ]], evaluated in the [[Riemann sphere|0CDMXGPSES]] with $a/0 = \\infty$ and $a/\\infty = 0$ for $a \\ne 0$.\nDefine $A_{-1} = 1$, $A_0 = b_0$, $B_{-1} = 0$, $B_0 = 1$ and, for $n \\ge 1$,\n$$A_n = b_n A_{n-1} + a_n A_{n-2}, \\qquad B_n = b_n B_{n-1} + a_n B_{n-2}.$$\nThen for every $n \\ge 0$:\n\n1. $x_n = \\dfrac{A_n}{B_n}$, where $A_n$ and $B_n$ are not both $0$;\n2. $A_n B_{n-1} - A_{n-1} B_n = (-1)^{n-1} a_1 a_2 \\cdots a_n$.\n\n## Remarks\n\nProof idea: write $x_n = t_0 \\circ t_1 \\circ \\cdots \\circ t_n(0)$ with $t_0(w) = b_0 + w$ and $t_k(w) = \\frac{a_k}{b_k + w}$.\nThese are [[Möbius transformations|0B346DDAM5]], and the product of their matrices is $\\begin{pmatrix} A_{n-1} \u0026 A_n \\\\ B_{n-1} \u0026 B_n \\end{pmatrix}$ by induction.\nTaking determinants gives item 2, and applying the product to $0$ gives item 1; item 2 shows that $A_n$ and $B_n$ do not both vanish.\nExample: for $1 + \\cfrac{1}{2 + \\cfrac{1}{2 + \\cdots}}$ the recurrences give $A_n = 1, 3, 7, 17, \\dots$ and $B_n = 1, 2, 5, 12, \\dots$, the convergents of $\\sqrt{2}$.\nItem 2 gives $x_n - x_{n-1} = \\dfrac{(-1)^{n-1} a_1 \\cdots a_n}{B_n B_{n-1}}$, so the convergents are the partial sums of a series, and convergence questions become questions about series.\n\n## Explanations\n\n### L1 human\n\nComputing a continued fraction from the inside out has to be redone each time a new level is added.\nThese formulas compute each new convergent from the two previous ones, with one multiplication and addition for its numerator and one for its denominator.\n\n### L2 human\n\nThe numerators $A_n$ and denominators $B_n$ satisfy a three-term linear recurrence, the same one for both.\nFor a [[simple continued fraction|06XF7BCFFJ]] they reduce to the recurrences for $p_n$ and $q_n$, and item 2 becomes $p_n q_{n-1} - p_{n-1} q_n = (-1)^{n-1}$.\nThe difference formula in the remarks is the starting point of the [[Śleszyński–Pringsheim theorem|NEC48RHNN0]] and other convergence tests for continued fractions.\n",
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