{
  "aliases": [
    "madhava-cosine-series",
    "madhava-sine-series",
    "madhavas-arctangent-series",
    "madhavas-cosine-series",
    "madhavas-series",
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  "body": "\n## Summary\n\nIn mathematics, the Maclaurin series of the sine, cosine, and arctangent functions are sometimes called **Madhava series** because they were discovered in the 14th or 15th century by the Indian mathematician Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are\n\n$$\\begin{alignedat}{3} \\sin \\theta \u0026= \\theta - \\frac{\\theta^3}{3!} + \\frac{\\theta^5}{5!} - \\frac{\\theta^7}{7!} + \\cdots \u0026\u0026= \\sum_{k=0}^\\infty \\frac{(-1)^k}{(2k+1)!}\\theta^{2k+1}, \\\\[10mu] \\cos \\theta \u0026= 1 - \\frac{\\theta^2}{2!} + \\frac{\\theta^4}{4!} - \\frac{\\theta^6}{6!} + \\cdots \u0026\u0026= \\sum_{k=0}^\\infty \\frac{(-1)^k}{(2k)!}\\theta^{2k}, \\\\[10mu] \\arctan x \u0026= x - \\frac{x^3}{3} + \\frac{x^5}{5} - \\frac{x^7}{7} + \\cdots \u0026\u0026= \\sum_{k=0}^\\infty \\frac{(-1)^k}{2k+1}x^{2k+1} \\quad \\text{where } |x| \\leq 1. \\end{alignedat}$$\n",
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  "name": "Madhava series",
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