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  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence such that the series $\\sum_{n=0}^{\\infty} a_n$ [[converges|0CVVKY7GY2]].\nLet $G(x) = \\sum_{n=0}^{\\infty} a_n x^n$ for $-1 \u003c x \u003c 1$.\nThen $G(x) \\to \\sum_{n=0}^{\\infty} a_n$ as $x \\to 1$ from the left.\n\n## Remarks\n\nThe [[power series|0VP56B4EMX]] converges on $(-1, 1)$, since its radius of convergence is at least $1$.\nThe theorem gives the sum of the [[alternating harmonic series|0JQ5Y3WWVE]]. Apply it to the [[Mercator series|0YZXNE3HFR]] for $\\ln(1 + x)$.\nThe converse fails. For $a_n = (-1)^n$, $G(x) = 1/(1 + x) \\to 1/2$, but $\\sum (-1)^n$ diverges.\nMathlib states it as `Real.tendsto_tsum_powerSeries_nhdsWithin_lt`.\n\n## Explanations\n\n### L1 human\n\nA power series that converges at the endpoint $x = 1$ is continuous there from the left.\nSo its value at $1$ is the limit of its values inside the interval.\n",
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  "name": "Abel's theorem",
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