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  "body": "\n## Summary\n\nAn asymptotic expansion of a function $f$ near a point is a formal series $\\sum a_n \\varphi_n$ in a scale of functions that shrink one after another.\nEvery partial sum must approximate $f$ with an error smaller than its last term.\nThe series need not converge.\nAsymptotic expansions describe special functions, integrals and solutions of differential equations for large or small parameters.\nThey are closely tied to the theory of [[divergent series|0NBRP6093V]] and to [[Borel summation|8DJPSYSHGJ]].\n\n## Explanations\n\n### L1 human\n\nThink of an approximation that gets better in stages: first a rough main term, then a correction, then a smaller correction.\nAn asymptotic expansion lists all these stages at once.\nEach stage is guaranteed to help when the variable is close enough to the limit point, even if adding infinitely many stages does not converge.\nFor example, as $x \\to \\infty$, the function $\\frac{1}{x - 1}$ has the expansion $\\frac{1}{x} + \\frac{1}{x^2} + \\frac{1}{x^3} + \\cdots$.\nThe [[Stirling series|K29QYWB6A5]] for $\\ln \\Gamma(z)$ is an expansion of this kind that diverges for every $z$ and is still very accurate.\n",
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