Radius of convergence of the binomial series

Statement

Let $\alpha$ be a real number that is not a natural number. Then the binomial series $\sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$ has radius of convergence $1$.

Remarks

The 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 gives $R = 1$. At $x = \pm 1$ the behavior depends on $\alpha$. For $\alpha > 0$ the series converges absolutely at both endpoints. Mathlib states the result as binomialSeries_radius_eq_one.

Explanations

L1 human

Unless the series stops after finitely many terms, it converges exactly inside $(-1, 1)$, plus possibly some endpoints. This matches the fact that $(1 + x)^{\alpha}$ has trouble at $x = -1$.