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  "body": "\n## Definition\n\nLet $f(x) \\sim \\sum_{n \\ge 0} a_n \\varphi_n(x)$ be an [[asymptotic expansion|BJQTEPEZ9G]] of $f$ as $x \\to x_0$, and fix a value of $x$ near $x_0$.\nAssume that the sequence $\\abs{a_n \\varphi_n(x)}$, $n \\ge 0$, attains a minimum, and let $N(x)$ be the smallest index $n$ at which it does.\nThe *optimally truncated* sum at $x$ is\n$$S_{\\mathrm{opt}}(x) = \\sum_{n=0}^{N(x) - 1} a_n \\varphi_n(x),$$\nthe partial sum that stops just before the smallest term.\n*Optimal truncation* is the approximation of $f(x)$ by $S_{\\mathrm{opt}}(x)$.\n\n## Remarks\n\nSome authors include the smallest term, or half of it; these choices change the error only by a fraction of the smallest term.\nThe index $N(x)$ depends on $x$ and usually tends to infinity as $x \\to x_0$.\n\nExample: $F(x) = \\int_0^\\infty \\frac{e^{-t}}{1 + xt}\\,dt \\sim \\sum_{n \\ge 0} (-1)^n n!\\, x^n$ as $x \\to 0^+$.\nFor $x = 0.1$ the terms $n!\\, (0.1)^n$ decrease while $n + 1 \u003c 10$, the terms with $n = 9$ and $n = 10$ are equal, and then they increase.\nSo $N(0.1) = 9$, and the error of the optimally truncated sum is at most the first omitted term $9!\\, (0.1)^9 \\approx 3.6 \\cdot 10^{-4}$, since the remainder after $N$ terms is bounded by the next term.\n\nNon-example: for a convergent series such as $\\frac{1}{1 - x} = \\sum x^n$ at $x = \\frac12$ the terms decrease forever, no minimum is attained, and optimal truncation is not defined; one simply sums the series.\n",
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