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  "body": "\n## Summary\n\nThe Rogers–Ramanujan continued fraction is $R(q) = \\cfrac{q^{1/5}}{1 + \\cfrac{q}{1 + \\cfrac{q^2}{1 + \\cdots}}}$ for $\\abs{q} \u003c 1$.\nIt equals $q^{1/5}$ times the quotient of the two series of the [[Rogers–Ramanujan identities|M24H4GJPPW]], hence an infinite product.\nAs a function of $\\tau$ with $q = e^{2\\pi i \\tau}$ it is a [[modular function|63TX60RTRQ]] for the group $\\Gamma(5)$.\nRamanujan found closed forms of its values at special points, such as $R(e^{-2\\pi}) = \\sqrt{\\frac{5 + \\sqrt5}{2}} - \\frac{1 + \\sqrt5}{2}$.\n\n## Explanations\n\n### L1 human\n\nA [[continued fraction|04P2RJ4WPS]] is a fraction whose denominator contains a fraction, whose denominator contains a fraction, and so on.\nIn this one the numerators are the powers $q, q^2, q^3, \\dots$ and the extra factor $q^{1/5}$ in front makes the function behave well under symmetries.\nAt $q = e^{-2\\pi}$ the value is an explicit number built from $\\sqrt5$, related to the [[golden ratio|0SC8T7548D]].\nRamanujan sent such evaluations to Hardy in his famous letters.\n",
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