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  "body": "\n## Statement\n\nLet $a_1, a_2, \\dots$ and $b_1, b_2, \\dots$ be complex numbers such that $\\abs{b_n} \\ge \\abs{a_n} + 1$ for every $n \\ge 1$.\nThen the [[continued fraction|0BK6JBK58W]]\n$$f = \\cfrac{a_1}{b_1 + \\cfrac{a_2}{b_2 + \\cfrac{a_3}{b_3 + \\cdots}}}$$\nconverges, and its value satisfies $\\abs{f} \\le 1$.\n\n## Remarks\n\nProof idea: with the [[fundamental recurrence formulas|6WYYMPH21Q]] ($b_0 = 0$), the denominators satisfy $\\abs{B_n} \\ge (\\abs{a_n} + 1)\\abs{B_{n-1}} - \\abs{a_n}\\abs{B_{n-2}}$.\nBy induction $\\abs{B_n} - \\abs{B_{n-1}} \\ge \\abs{a_1 \\cdots a_n}$, so $1 = \\abs{B_0} \\le \\abs{B_1} \\le \\cdots$ and no $B_n$ vanishes.\nThen $\\abs{f_n - f_{n-1}} = \\frac{\\abs{a_1 \\cdots a_n}}{\\abs{B_n}\\abs{B_{n-1}}} \\le \\frac{1}{\\abs{B_{n-1}}} - \\frac{1}{\\abs{B_n}}$, and this telescopes: the series $\\sum (f_n - f_{n-1})$ converges absolutely with total at most $1$, where $f_0 = 0$.\nExample: $a_n = 1$ and $b_n = 2$ satisfy the condition, and the value is $\\sqrt{2} - 1$.\nThe condition cannot simply be dropped: for $a_n = -1$ and $b_n = 1$ one has $\\abs{b_n} \u003c \\abs{a_n} + 1$, and the convergents repeat the cycle $-1, \\infty, 0$, so the continued fraction diverges.\n\n## Explanations\n\n### L1 human\n\nA continued fraction converges when each partial denominator is larger than the partial numerator by at least one in absolute value.\nThe value then lies in the closed unit disc.\n\n### L2 human\n\nThis is a basic sufficient condition for convergence, in the same spirit as the comparison test for series.\nEach map $w \\mapsto \\frac{a_n}{b_n + w}$ sends the closed unit disc into itself under the hypothesis, which explains the bound $\\abs{f} \\le 1$.\nThe [[Van Vleck theorem|KCCFZTP3MM]] is another classical criterion, for partial denominators in a sector.\n",
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    "name": "Śleszyński–Pringsheim theorem",
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      "wikipedia": "Śleszyński–Pringsheim theorem"
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    "tag": "NEC48RHNN0",
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  "msc": [
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  "name": "Śleszyński–Pringsheim theorem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/NEC/NEC48RHNN0.md",
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