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  "body": "\n## Summary\n\nA Poincaré asymptotic series of a function $f$ at infinity is a formal power series $\\sum a_n z^{-n}$ with one property.\nEvery partial sum approximates $f$ with an error smaller than the last power used.\nIt is the [[asymptotic expansion|BJQTEPEZ9G]] for the [[asymptotic scale|XEGB8Y44GG]] of the powers $z^{-n}$, as defined by Poincaré in 1886.\nThe series may diverge for every $z$, and still give very accurate approximations.\nThe coefficients are unique when they exist, but different functions can have the same series.\n\n## Explanations\n\n### L1 human\n\nThink of a series in powers of $1/z$ that is useful for large $z$ even though it may never converge.\nThe rule is: stop after any finite number of terms, and the error goes to zero faster than the last term you kept.\nFor example, $\\frac{1}{z - 1} = \\frac1z + \\frac{1}{z^2} + \\frac{1}{z^3} + \\cdots$ is a Poincaré series at infinity that happens to converge.\nThe series $\\sum_{n \\ge 0} (-1)^n n!\\, z^{-n-1}$ of the [[exponential integral|EV6ZDQ0NDY]] $e^z E_1(z)$ is one that diverges for every $z$.\n",
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