{
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  "body": "\n## Statement\n\nFor every real $x$ with $|x| \\le 1$,\n$$\\arctan x = \\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n+1}}{2n + 1} = x - \\frac{x^3}{3} + \\frac{x^5}{5} - \\cdots.$$\n\n## Remarks\n\nFor $|x| \u003c 1$, integrate the [[geometric series|01X5CVV1Q5]] $\\frac{1}{1 + t^2} = \\sum (-1)^n t^{2n}$ term by term from $0$ to $x$. Its sum is the [[derivative of arctangent|05P4B4Q22V]].\nAt $x = \\pm 1$, the series converges by the alternating series test, and [[Abel's theorem|0S4EMDQS2F]] gives the equality. At $x = 1$ this is the [[Leibniz formula|037FD5NBRT]] $\\pi/4 = 1 - 1/3 + 1/5 - \\cdots$.\nThe radius of convergence is $1$. Mathlib states the series for $|x| \u003c 1$ as `Real.hasSum_arctan`.\n\n## Explanations\n\n### L1 human\n\nThe arctangent is a power series with odd powers and the simple coefficients $\\pm 1/(2n+1)$.\nAt $x = 1$, it gives a famous series for $\\pi/4$.\n",
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