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  "body": "\n## Statement\n\nLet $\\sum_{n=0}^{\\infty} a_n (x - c)^n$ be a [[power series|0VP56B4EMX]] with [[radius of convergence|0660QX1VPR]] $R \u003e 0$.\nThen for every $r$ with $0 \u003c r \u003c R$, the series converges absolutely and uniformly on $[c - r, c + r]$.\n\n## Remarks\n\nChoose $s$ with $r \u003c s \u003c R$. The series converges at $c + s$, so $|a_n| s^n \\le C$ for some $C$. The [[M-test|0NX5NWAFC9]] with $M_n = C (r/s)^n$ applies.\nSo the sum is continuous on $(c - R, c + R)$.\nThe convergence need not be uniform on the whole interval: $\\sum x^n$ is not uniformly convergent on $(-1, 1)$. Mathlib states the result as `HasFPowerSeriesOnBall.tendstoUniformlyOn`.\n\n## Explanations\n\n### L1 human\n\nInside the interval of convergence, a power series converges uniformly once we stay away from the endpoints.\nNear the endpoints, the convergence can get slower and slower.\n",
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