= Optimal truncation
<Optimal truncation> chooses how many terms to retain for a useful finite approximation. In a divergent <asymptotic expansion>, increasing the truncation order eventually makes the terms larger rather than smaller. Stopping near a smallest term often gives the best direct approximation, but the actual remainder must be checked. A small last term alone is not a universal <error bound>.<NIST Digital Library of Mathematical Functions/2.11 Remainder Terms; Stokes Phenomenon>[NIST Digital Library of Mathematical Functions, §2.11(i), discussion of finite-parameter accuracy and independently checking remainders; §2.11(iii), exponentially improved expansions]{ref}
For <a divergent factorial series can approximate a convergent integral>, $T_k=k!/z^k$ decreases while $k+1<z$ and increases while $k+1>z$. Thus the useful cutoff is near $k=z$, rather than at arbitrarily large order. At $z=10$, $T_9=T_{10}=0.00036288$ are the two smallest terms. Retaining $N=10$ terms gives
$$
S_{10}(10)=0.91545632,\qquad
|F(10)-S_{10}(10)|\leq0.00036288.
$$
The actual error is about $0.0001770$. Retaining twenty terms instead gives $S_{20}(10)\approx0.9076137084$, with error about $0.008020$: more work produces a worse answer.
\Image[/asymptotic-factorial-truncation-error.png]
{title=A divergent <asymptotic series> has a useful stopping region}
{description=Original numerical errors $|F(z)-S_N(z)|$ for $z=4,8,10,16$ and $1\leq N\leq26$. Each <partial sum> is compared with an independent evaluation of the positive <integral>. The dots mark the smallest errors among the displayed orders; they are not a claim of a universal stopping theorem.}
For this example the bound $N!/z^N$ is explicit, so a requested tolerance can be tested before using a truncation. For a different <integral>, a nearby <saddle point> or <simple pole> can change the relevant remainder scale. This is why the smallest-term rule is a useful guide, rather than a replacement for remainder analysis.
The cutoff for factorial terms $n!z^{-n}$ has $n=|z|+O(1)$. Better analysis can expose contributions smaller than every retained inverse power, linking <optimal truncation> to the <Stokes phenomenon>. Such contributions can be important even when many asymptotic <coefficients> agree. Directly summing ever more terms does not automatically recover them.
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