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  "body": "\n## Statement\n\nThe [[series|0AF7X08PCR]] below converges, and\n$$\\frac{1}{\\pi} = \\frac{2\\sqrt2}{9801}\\sum_{k=0}^{\\infty} \\frac{(4k)!\\,(1103 + 26390k)}{(k!)^4\\,396^{4k}},$$\nwhere $\\pi$ is the number [[pi|0C1VYVNV2N]] and $n!$ is the [[factorial|0XWG0JGDFW]].\n\n## Remarks\n\nConvergence: by Stirling's formula $(4k)!/(k!)^4$ grows like a constant times $256^k k^{-3/2}$, while $396^4 \\approx 2.46 \\times 10^{10}$, so the ratio of successive terms tends to $256/396^4 \\approx 1.04 \\times 10^{-8}$.\nSo each term adds about eight correct decimal digits.\nExample: the $k = 0$ term alone gives $\\pi \\approx 9801/(2\\sqrt2 \\cdot 1103) = 3.14159273\\ldots$, against $\\pi = 3.14159265\\ldots$.\nRamanujan published this series in 1914 among several series of the same shape, without a complete proof.\nProof idea: the series comes from the theory of [[complete elliptic integrals|RM54CPPE12]] and modular equations.\nIt uses a singular value of the modulus attached to the imaginary quadratic field $\\Q(\\sqrt{-58})$, which is reflected in $e^{\\pi\\sqrt{58}} \\approx 396^4 - 104$.\nIdentities such as [[Legendre's relation|8GSKDNZ4TK]] convert the resulting relations between elliptic integrals into a formula for $1/\\pi$.\nA complete proof was given later by Jonathan and Peter Borwein.\n\n## Explanations\n\n### L1 human\n\nThis is a formula that writes $1/\\pi$ as an infinite sum of fractions built from factorials and powers of $396$.\nIts terms shrink so fast that each new term gives about eight more digits of $\\pi$.\nThe first term alone already gives $\\pi$ to six decimal places.\n\n### L2 human\n\nThe series is the most famous member of the family of Ramanujan–Sato series, in which $1/\\pi$ equals a sum of hypergeometric-type terms $\\frac{(4k)!}{(k!)^4}(a + bk)x^k$ or similar.\nEach such series comes from a singular value of the modulus, where the elliptic curve has complex multiplication.\nThe same idea with the $j$-invariant and the field $\\Q(\\sqrt{-163})$ gives the [[Chudnovsky algorithm|JKCKXDT5B5]], which gains about fourteen digits per term.\n",
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