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  "body": "\n## Summary\n\nOptimal truncation is the rule of stopping a divergent [[asymptotic expansion|BJQTEPEZ9G]] just before its smallest term, for a fixed value of the variable.\nFor many series of factorial growth the error at this point is about the size of the smallest term.\nThis size is exponentially small in the large parameter.\nSo a divergent series can still give an approximation of very high accuracy.\nThe exponentially small error left after optimal truncation is the starting point of hyperasymptotics and of the theory of [[resurgent functions|37R4QJCARA]].\n\n## Explanations\n\n### L1 human\n\nIn a divergent asymptotic series the terms first get smaller and then start to grow again.\nAdding terms helps while they shrink and hurts once they grow.\nSo the best one can do is to stop at the smallest term.\nFor example, for the series $1 - x + 2x^2 - 6x^3 + \\cdots$ with terms $n!\\, x^n$ and $x = 0.1$, the terms are smallest near $n = 10$, and stopping there gives an error of about $2 \\cdot 10^{-4}$.\n",
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