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  "body": "\n## Statement\n\nLet $\\alpha$ be a real number that is not a natural number.\nThen the [[binomial series|0HB0XKDXHH]] $\\sum_{k=0}^{\\infty} \\binom{\\alpha}{k} x^k$ has radius of convergence $1$.\n\n## Remarks\n\nThe coefficients are nonzero, and $\\binom{\\alpha}{k+1} / \\binom{\\alpha}{k} = (\\alpha - k)/(k + 1)$. Its absolute value tends to $1$, so the [[ratio formula|0JBMYAA9QH]] gives $R = 1$.\nAt $x = \\pm 1$ the behavior depends on $\\alpha$. For $\\alpha \u003e 0$ the series converges absolutely at both endpoints.\nMathlib states the result as `binomialSeries_radius_eq_one`.\n\n## Explanations\n\n### L1 human\n\nUnless the series stops after finitely many terms, it converges exactly inside $(-1, 1)$, plus possibly some endpoints.\nThis matches the fact that $(1 + x)^{\\alpha}$ has trouble at $x = -1$.\n",
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