Optimal Truncation and Superasymptotics

A Poincaré asymptotic series $f(z)\sim\sum_{n=0}^{\infty}a_n z^{-n}$ is built so that each finite partial sum approximates $f$ with error $o(z^{-N})$ as $z\to\infty$ . The definition says nothing about the limit $N\to\infty$ at fixed $z$ — and for the generic series thrown up by physics that limit does not exist: the coefficients grow factorially and the series diverges for every $z$ . Yet the partial sums are not useless: as you add terms they first march toward the true value, reach a best accuracy, and only then run away. Optimal truncation answers the question the definition dodges: at fixed $z$ , how many terms should you add? The rule is disarmingly simple: stop at the smallest term. The error made by truncating at order $N$ is comparable to the first omitted term, so the tightest approximation sits at $N=N_\star$ , the index of the least term. For the universal case of factorial divergence $a_n\sim C\,n!/A^n$ , the least term lands at $N_\star\approx A|z|$ and, by Stirling, has magnitude $\sim e^{-A|z|}$ . An optimally truncated series is therefore accurate to a floor exponentially small in $|z|$ — smaller than every algebraic power $z^{-N}$ . Michael Berry named this regime, beyond all orders of the original power series, superasymptotics. That exponentially small floor is not numerical noise. It is a genuine second exponential — a tunneling amplitude, a subdominant saddle, an instanton — that switches on across a Stokes line and is captured by hyperasymptotics, Borel summation, and resurgence. Optimal truncation is thus both the practical endpoint of elementary asymptotics and the doorway to the exponential asymptotics beyond it, underlying error control in Watson's lemma, steepest descent, and WKB.

Key takeaways

A divergent asymptotic series $\sum_n a_n z^{-n}$ never converges, yet its partial sums first approach $f$ , reach a best accuracy, then diverge — so "add more terms" is wrong past a point.
The truncation error is roughly the size of the first omitted term, so the optimal order $N_\star$ is where the terms are smallest: the least term.
For factorial coefficients $a_n\sim n!/A^n$ the optimal order is $N_\star\approx A|z|$ and the minimal error is exponentially small, $\varepsilon_{\min}\sim\sqrt{2\pi A|z|}\,e^{-A|z|}$ .
This accuracy beyond all algebraic orders is superasymptotics; the exponentially small remainder is a real physical effect, not roundoff.
Pushing below the floor needs hyperasymptotics, Borel summation, and resurgence — the divergent tail must be acknowledged, not ignored.
Key idea: stop summing a divergent asymptotic series at the smallest term — that optimal truncation minimizes the error, and for factorial divergence the minimal error is exponentially small, $\sim e^{-A|z|}$ .

The core insight

Stop summing a divergent asymptotic series at the smallest term: the optimal truncation order minimizes the error, and for factorial divergence that minimal error is exponentially small, $\sim e^{-A|z|}$ .

Where this lives today

Optimal truncation is quietly everywhere. Scientific libraries evaluate the gamma, Bessel, and error functions for large argument by summing their divergent asymptotic series to the least term — faster and more accurate than any convergent alternative. In quantum mechanics the WKB series in $\hbar$ diverges, and its least-term floor is exactly the tunneling amplitude $e^{-S_0/\hbar}$ : the remainder that superasymptotics exposes is the nonperturbative barrier-penetration effect, tied to Bohr–Sommerfeld corrections. In quantum field theory the perturbation series diverge factorially, with $A$ set by instanton or renormalon actions, and optimal truncation gives the best purely perturbative estimate before Borel–Écalle resurgence takes over — a program now central to gauge theories. The same mathematics governs the exponentially small splitting of separatrices in chaotic dynamics and dendritic crystal growth, wherever a "beyond all orders" effect, invisible to every term of a power series, decides the physics.

The physical picture

Line up the magnitudes of the terms, $t_n=|a_n z^{-n}|$ , and watch them as $n$ climbs. At first the power-law suppression $z^{-n}$ wins and each new term is smaller than the last — a genuine correction. But the coefficients $a_n$ grow factorially, and factorial growth eventually overwhelms any fixed power of $z$ , so the terms bottom out and then climb again: the sequence $t_n$ is a shallow bowl. While the bowl descends every term earns its keep; once it rises every term is worse than the error you already had. The bottom of the bowl is where to stop, and its depth is the smallest error the series can deliver.

Figure

The term magnitudes tₙ = n!/10ⁿ for the Euler series at z = 10 form a shallow bowl: they fall to a least term of about 3.6×10⁻⁴ near n = 9 to 10, then climb without bound. The optimal truncation order is N⋆ ≈ Az = 10 and the depth of the bowl is the superasymptotic error floor

Why should the least term set the error? For a well-behaved (say alternating) series the true value is trapped between consecutive partial sums, and the gap between them is exactly one term; the narrowest gap is at the least term, the tightest bracket you can throw around $f$ . Factorial growth $a_n\sim n!/A^n$ is the universal fingerprint of series born from saddle points and perturbation theory — the $n!$ counts the proliferating ways an expansion can go wrong, singularities in the Borel plane — and Stirling turns "least term" into "error $\sim e^{-A|z|}$ ", so the geometry of divergence directly encodes an exponentially small physical effect. That exponentially small remainder does something, and it does it abruptly. As an external phase is swept, the hidden second exponential $e^{-A|z|}$ switches on — its coefficient jumps from $0$ to $1$ across a Stokes line. Berry's discovery is that the jump is not really discontinuous: at finite $|z|$ it is smeared into a universal error function of width $\sim(A|z|)^{-1/2}$ , sharpening to the classical discontinuity only as $|z|\to\infty$ .

Figure

Berry's universal smoothing of the Stokes jump: as the phase θ sweeps across a Stokes line the subdominant exponential e^(-Az) is switched on smoothly, its multiplier σ rising from 0 to 1 as an error function of width Δθ ∼ (Az)^(-1/2). The value 1/2 sits exactly on the line; the sharp classical discontinuity is the z → ∞ limit

The core equations

A Poincaré asymptotic expansion is defined by its error bound at each fixed order, not by convergence. The relation below is the object of study: a formal power series in $z^{-1}$ whose coefficients typically grow like $n!$ , so it diverges for every finite $z$ even though it faithfully approximates $f$ order by order.

f(z)\ \sim\ \sum_{n=0}^{\infty} a_n\,z^{-n},\qquad z\to\infty \qquad (1)

where: $f(z)$ — the function being approximated, usually an integral or special function; $a_n$ — asymptotic coefficients, typically growing like $n!$ ; $z$ — the large asymptotic parameter. Reading: each partial sum is good to $o(z^{-N})$ at fixed $N$ , but summing all terms is meaningless. Truncating at order $N$ leaves a remainder. The most useful fact in the subject is that this remainder is controlled by the first term you threw away — the error is essentially the size of term $N+1$ .

R_N(z)\equiv f(z)-\sum_{n=0}^{N} a_n z^{-n}=O\!\big(a_{N+1}z^{-(N+1)}\big) \qquad (2)

where: $R_N(z)$ — remainder after summing through order $N$ ; $a_{N+1}z^{-(N+1)}$ — the first neglected term, which sets the error scale. Reading: to minimize the error, minimize the first omitted term. To make "the smallest term" precise, track the magnitude $t_n$ of each term and define the optimal order as the index that minimizes it — the rule of optimal truncation in two symbols.

t_n(z)=\big|a_n z^{-n}\big|,\qquad N_\star(z)=\arg\min_{n}\, t_n(z) \qquad (3)

where: $t_n$ — magnitude of the $n$ -th term; $N_\star$ — the truncation order that minimizes it and hence the error. Reading: $N_\star$ is where the bowl in the first figure bottoms out. Locating that minimum is a one-line calculation: the least term sits where consecutive terms stop shrinking and start growing, where the term ratio passes through one. For factorial coefficients this pins the optimal order to grow linearly with $|z|$ .

\frac{t_{n+1}}{t_n}=\frac{|a_{n+1}|}{|a_n|}\,|z|^{-1}\approx 1\quad\Longrightarrow\quad N_\star\approx A\,|z| \qquad (4)

where: $A$ — the radius of the nearest Borel-plane singularity, setting the divergence rate through $a_n\sim n!/A^n$ . Reading: doubling $z$ lets you keep about twice as many useful terms. Evaluating the least term with Stirling's formula gives the punchline: the powers of $z$ cancel, leaving a clean exponential — the best accuracy a single optimally truncated series can achieve.

\varepsilon_{\min}=t_{N_\star}\sim \sqrt{2\pi A|z|}\;e^{-A|z|} \qquad (5)

where: $\varepsilon_{\min}$ — the minimal achievable error, the depth of the bowl. Reading: an algebraic power series delivers exponential accuracy, but no better, on its own — the superasymptotic floor. Packaged as an approximation with an explicit error term, optimal truncation states that $f$ equals its least-term partial sum up to a remainder smaller than every power of $z^{-1}$ .

f(z)=\sum_{n=0}^{N_\star} a_n z^{-n}\;+\;\mathcal{O}\!\big(e^{-A|z|}\big) \qquad (6)

where: $\mathcal{O}(e^{-A|z|})$ — the exponentially small superasymptotic remainder, where the physics hides. To break through the floor you re-expand the remainder itself in a second divergent series times the subdominant exponential, with a Stokes multiplier that switches it on. Iterating is hyperasymptotics.

f(z)=\sum_{n=0}^{N_\star} a_n z^{-n}\;+\;\sigma(z)\,e^{-A z}\sum_{m} b_m z^{-m}+\cdots \qquad (7)

where: $\sigma(z)$ — the Stokes multiplier, the smooth switch-on of the subdominant exponential; $b_m$ — coefficients of the re-expanded series. Reading: each stage buys a further slice of the exponent. Finally, the whole tower is organized by the Borel transform, whose nearest singularity at $t=A$ is the origin of the factorial growth throughout.

f(z)=\int_0^{\infty} B(t)\,e^{-z t}\,\mathrm{d}t,\qquad B(t)=\sum_{n} \frac{a_n}{(n-1)!}\,t^{\,n-1} \qquad (8)

where: $B(t)$ — the Borel transform; its singularity at $t=A$ explains $a_n\sim n!/A^n$ and reconstructs $f$ exactly. Reading: divergence in $z$ -space is a singularity in Borel space.

Derivation ALocating the least term and the superasymptotic floor

The optimal truncation order and the size of the error floor both fall out of one elementary optimisation: follow the magnitude of successive terms, find where the ratio of consecutive terms crosses one, and evaluate the smallest term with Stirling's formula. The entire content of superasymptotics is hidden here, where the factorial growth of the coefficients fights the power-law suppression of $z^{-n}$ until the two balance at the least term.

Step 1. Form the ratio of successive term magnitudes, $\dfrac{t_{n+1}}{t_n}=\dfrac{|a_{n+1}|}{|a_n|\,|z|}$ . The terms decrease while this ratio is below one and increase once it exceeds one, so the least term sits where the ratio is approximately one. Why: a minimum of a positive sequence is exactly the point where it stops falling and starts rising, which for a product like $t_n$ is cleanest to detect through the ratio.

Step 2. Insert the universal factorial growth $a_n\sim n!/A^n$ , so that $\dfrac{|a_{n+1}|}{|a_n|}=\dfrac{(n+1)!/A^{n+1}}{n!/A^n}=\dfrac{n+1}{A}$ . The coefficient ratio grows linearly in $n$ , and this growth is what forces the terms to turn around. Why: only a coefficient ratio that grows without bound can overcome the fixed suppression $1/|z|$ and force divergence; a bounded ratio would give convergence.

Step 3. Set the term ratio to one: $\dfrac{n+1}{A|z|}=1\Rightarrow N_\star\approx A|z|-1\approx A|z|$ . The optimal order grows linearly with the large parameter, so larger $z$ means more usable terms and a deeper bowl. Why: this is the crossover point identified in Step 1, now made quantitative — it is the location of the bottom of the term-magnitude bowl.

Step 4. Evaluate the least term. Substitute $n=N_\star\approx A|z|$ into $t_{N_\star}=\dfrac{N_\star!}{A^{N_\star}|z|^{N_\star}}$ and apply Stirling $N_\star!\sim\sqrt{2\pi N_\star}(N_\star/e)^{N_\star}$ :

t_{N_\star}\sim\sqrt{2\pi A|z|}\,\frac{(A|z|/e)^{A|z|}}{(A|z|)^{A|z|}}=\sqrt{2\pi A|z|}\;e^{-A|z|}.

The two towers of $(A|z|)^{A|z|}$ cancel exactly, leaving a bare exponential times an algebraic prefactor. Why: the cancellation of the powers is the mathematical heart of superasymptotics — it is precisely why the floor is exponentially, not merely algebraically, small. *Verified: term ratio $(n+1)/A|z|=1$ gives $N_\star\approx A|z|$ ; Stirling gives $\varepsilon_{\min}\sim\sqrt{2\pi A|z|}\,e^{-A|z|}$ ; for the Euler series ( $A=1$ ) at $z=10$ this predicts $N_\star=10$ and floor $\approx 3.6\times10^{-4}$ , matching the tabulated least term below.*

Go deeper · Derivation B · 2 minWhy the error is the size of the first omitted term

Derivation BWhy the error is the size of the first omitted term

For a strictly alternating asymptotic series whose terms first decrease, the remainder is not merely bounded by the first omitted term — it is comparable to it, which is what makes the least term a genuine error estimate rather than a loose bound. We establish the alternating-series bracket, read off that the gap between consecutive partial sums equals one term, and conclude that the tightest bracket, and hence the smallest error, sits at the least term — the same estimate that serves as the working heuristic for generic factorial series.

Step 1. For an alternating series with decreasing terms the partial sums straddle the true value, $S_N<f<S_{N+1}$ or the reverse, where $S_N=\sum_{n=0}^N a_n z^{-n}$ : each new term overshoots on the opposite side, so consecutive sums bracket $f$ . Why: alternating signs with shrinking magnitudes is the classical Leibniz situation, and bracketing turns partial sums into a rigorous confidence interval for $f$ .

Step 2. The width of that bracket is one term: $|f-S_N|=|R_N|\le|a_{N+1}z^{-(N+1)}|=t_{N+1}$ . The remainder is bounded by — and, since the true value sits near the middle of the bracket, comparable to — the first omitted term. Why: this converts the abstract remainder $R_N$ into something you read straight off the terms you already computed, with no knowledge of $f$ required.

Step 3. Minimize the bound over $N$ . Since $t_{N+1}$ inherits the bowl shape of $t_n$ , the tightest bracket is at $N=N_\star$ : $\dfrac{\mathrm{d}}{\mathrm{d}N}t_N=0\Rightarrow N=N_\star$ , giving $\min_N|R_N|\approx t_{N_\star}$ . Why: choosing $N$ to minimize the guaranteed error is the definition of optimal truncation — the narrowest available bracket around $f$ .

Step 4. Insert the factorial growth from the previous derivation to get the explicit floor $\big|R_{N_\star}\big|\approx t_{N_\star}\sim\sqrt{2\pi A|z|}\,e^{-A|z|}$ . A useful refinement: because $f$ lies between $S_{N_\star}$ and $S_{N_\star+1}$ , their average cancels the leading remainder and beats either endpoint. Why: the midpoint trick is a free half-order of accuracy and the seed of the hyperasymptotic idea that the remainder itself has structure worth resolving. *Verified: alternating bracket $S_9<f<S_{10}$ for the Euler series at $z=10$ gives $0.915456<0.915633<0.915819$ , an optimal error $\approx1.8\times10^{-4}$ (about half the least term), and midpoint $0.915638$ versus true $0.915633$ .*

Go deeper · Derivation C · 1 minFrom the least term to the Stokes jump

Derivation CFrom the least term to the Stokes jump

The exponentially small remainder that optimal truncation uncovers is not numerical dust — it is a second, subdominant exponential $e^{-Az}$ representing a real object such as a tunneling amplitude or a subordinate saddle, whose coefficient switches on across a Stokes line. The goal here is to see, at the level of scaling, why the remainder carries the factor $e^{-Az}$ , why the switch-on happens at the least-term order, and why Berry's smoothing turns the classical discontinuous jump into a universal error function of width $|z|^{-1/2}$ .

Step 1. Write the optimally truncated remainder as a Borel-tail integral, $R_N(z)\propto z^{-N}\!\int_0^\infty B^{(N)}(t)\,e^{-zt}\,\mathrm{d}t$ , dominated by the Borel transform near its nearest singularity at $t=A$ . Why: recasting the leftover sum as an integral exposes the analytic structure — the singularity at $t=A$ — that the divergent power series only hinted at through its factorial coefficients.

Step 2. The integrand peaks where $e^{-zt}$ and the singular growth of $B^{(N)}$ balance, at $t\approx A$ , which coincides with the least-term order $N\approx A|z|$ , giving a contribution of order $e^{-Az}$ . Why: the least term and the Borel singularity live at the same scale $A$ — the place where the series is optimally truncated is the place where the hidden exponential is born.

Step 3. Track the coefficient of $e^{-Az}$ as the phase of $z$ crosses the Stokes line. Berry's result is that it rises smoothly, $R_{N_\star}\approx\tfrac12\,\mathrm{erfc}\!\big(\theta\,\sqrt{A|z|/2}\big)\,e^{-Az}$ , from $0$ to $1$ over a width $\Delta\theta\sim(A|z|)^{-1/2}$ , passing through exactly $\tfrac12$ on the line. Why: the discontinuous jump in the multiplier is only the $|z|\to\infty$ limit; at finite $|z|$ the switch-on is an error function — the content of the second figure, and the reason the Stokes phenomenon is smooth. *Verified: floor scale $e^{-A|z|}$ coincides with the Borel-singularity location $t=A$ and the least-term order $N_\star=A|z|$ ; the erfc smoothing follows Berry (1989) and Berry–Howls (1990). Full derivation belongs to the Stokes-phenomenon and Borel-summation articles.*

Limiting cases

How the optimal order and the error floor depend on the growth rate of the coefficients (here $z>0$ large):

Case	Result	Meaning
Factorial (Gevrey-1)	$a_n\sim C\,\dfrac{n!}{A^{n}}\Rightarrow N_\star\approx A z,\ \varepsilon_{\min}\sim e^{-A z}$	The generic case from steepest descent, Watson's lemma, and perturbation theory; the floor is a clean exponential.
Factorial-squared (Gevrey-2)	$a_n\sim C\,\dfrac{(n!)^2}{A^{n}}\Rightarrow N_\star\approx \sqrt{A z},\ \varepsilon_{\min}\sim e^{-2\sqrt{A z}}$	Faster (double-instanton-type) divergence gives fewer usable terms and a weaker, stretched-exponential floor.
Geometric (convergent)	$a_n\sim C\,r^{n}\ (r<1)\Rightarrow$ terms always decrease	No least term in range: the series converges, so truncation can be pushed arbitrarily far — optimal truncation is moot.
Borel-summable tail	$f=(\text{opt. trunc.})+\underbrace{\mathcal{O}(e^{-A z})}_{\text{recovered by Borel sum}}$	Below the floor: the remainder is reconstructed exactly by Borel summation, restoring the full function.

Worked examples

1 · The Euler series at z = 10 (fully worked)

The function $f(z)=\int_0^\infty\dfrac{e^{-t}}{1+t/z}\,\mathrm{d}t=z\,e^{z}E_1(z)$ has the divergent Euler asymptotic series $\sum_n(-1)^n n!\,z^{-n}$ , so $a_n=(-1)^n n!$ and $A=1$ . Truncate optimally at $z=10$ and compare with the exact value $f(10)=0.915633$ .

Step 1. Locate the optimal order. With $a_n=(-1)^n n!$ the term ratio is $t_{n+1}/t_n=(n+1)/z$ , which equals one at $n+1=z$ , so $N_\star\approx z=\mathbf{10}$ . The terms $t_n=n!/10^n$ run $1,\,0.1,\,0.02,\,6.0\times10^{-3},\dots$ down to a least term $t_9=t_{10}=\mathbf{3.63\times10^{-4}}$ , then climb again — the bowl of the first figure.

Step 2. Compare with the Stirling floor. Equation (5) predicts $\varepsilon_{\min}\sim\sqrt{2\pi\cdot10}\,e^{-10}=7.93\times4.54\times10^{-5}=\mathbf{3.60\times10^{-4}}$ , matching the tabulated least term to two figures. Note the two faces of the floor: the bare exponential order $e^{-10}=4.5\times10^{-5}$ , dressed by the prefactor $\sqrt{2\pi\cdot10}\approx7.9$ to give the realized $3.6\times10^{-4}$ .

Step 3. Sum to the least term. The alternating partial sums bracket the true value: $S_9=0.915456$ and $S_{10}=0.915819$ straddle $f(10)=0.915633$ , so the optimal error is $|f-S_9|=\mathbf{1.8\times10^{-4}}$ — about half the least term, exactly as the alternating bound predicts. Averaging the brackets gives $\tfrac12(S_9+S_{10})=\mathbf{0.915638}$ , correct to $5\times10^{-6}$ : a free extra order from the midpoint trick.

Step 4. Watch it fail past the floor. Keep going and the terms grow: $t_{15}=1.3\times10^{-3}$ , $t_{20}=2.4\times10^{-2}$ , so $S_{20}$ is worse than $S_{10}$ by two orders of magnitude. Stopping at $N_\star=10$ gives error $1.8\times10^{-4}$ ; pressing on to $N=20$ gives error $\sim10^{-2}$ — the least term is the best this series can do.

2 · Stirling's series for the Gamma function

Stirling's asymptotic series $\ln\Gamma(z)\sim(z-\tfrac12)\ln z-z+\tfrac12\ln2\pi+\sum_{n\ge1}\dfrac{B_{2n}}{2n(2n-1)z^{2n-1}}$ diverges because the Bernoulli numbers grow like $B_{2n}\sim(-1)^{n-1}\dfrac{2(2n)!}{(2\pi)^{2n}}$ . Find where to truncate at $z=10$ and how accurate you can get.

Step 1. Identify the divergence rate. The nearest Borel singularity sits at $2\pi$ (the first pole of the Bernoulli generating function), so the growth constant is $A=2\pi$ — much larger than the Euler series' $A=1$ , pushing the least term deeper and further out.

Step 2. Optimal order. Balancing $(2n)!$ against $(2\pi z)^{2n}$ gives the least term near $2n\approx2\pi z$ , i.e. $N_\star\approx\pi z$ . For $z=10$ that is $N_\star\approx\mathbf{31}$ terms.

Step 3. Error floor. The optimally truncated remainder is of order $e^{-2\pi z}$ . At $z=10$ this is $e^{-20\pi}=e^{-62.8}\approx\mathbf{5\times10^{-28}}$ — Stirling's formula, cut at its least term, pins $\ln\Gamma(10)$ to nearly thirty digits. The same optimal-truncation rule that gives $10^{-4}$ for the Euler series gives $10^{-28}$ here, purely because $A=2\pi$ instead of $A=1$ — the floor $e^{-A z}$ is exponentially sensitive to the Borel radius.

Physics of the portrait

Optimal truncation is old craft; the theory of why it works, and what lies beneath it, runs from Stokes's puzzlement in the 1850s to Berry's smoothing in the 1980s.

Origins · 1 minMichael Berry and how this was discovered

Michael Berry

b. 1941 · exponential asymptotics and the Stokes phenomenon

Michael Victor Berry, at Bristol, is best known for the geometric phase that bears his name, but his work on exponential asymptotics is what turned the folklore of optimal truncation into a precise theory. The lineage is long. George Gabriel Stokes, computing the asymptotics of the Airy function in 1857, noticed that the constants multiplying subdominant exponentials seemed to jump discontinuously as the phase of the argument changed — the "Stokes phenomenon" — but could not say why. Poincaré gave asymptotic series their rigorous definition in 1886, and Stieltjes in the same year analysed the archetypal divergent series carrying his name, establishing the least-term bound. For most of the twentieth century the divergence was treated as an embarrassment; Robert Dingle changed the tone in 1973, insisting that the divergent tail be interpreted, not discarded, and introducing the "terminants" that resum it. Berry's decisive step came in 1989: he showed that Stokes's discontinuous jump is, at finite parameter, a smooth error function of universal width $\sim|z|^{-1/2}$ , and with John Howls he named the two regimes beyond ordinary asymptotics — superasymptotics, the exponentially accurate floor of one optimally truncated series, and hyperasymptotics, the systematic descent below it. Jean Écalle's resurgence (1981) and John Boyd's 1999 survey completed the synthesis: the least term is a doorway, and the exponentially small remainder is a physical object with a life of its own.

History

1857

— Stokes traces his exponential discontinuities in the asymptotics of the Airy function

1886

— Poincaré gives the rigorous definition of an asymptotic power series

1886

— Stieltjes analyses the divergent series that now bears his name

1973

— Dingle systematises the interpretation of divergent-expansion terminants

1989

— Berry smooths the Stokes discontinuity into a universal error function

Going further

Texts. Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers, Ch. 3 (asymptotic series, least term, Stieltjes and Euler series) · Stone & Goldbart, Mathematics for Physics, chapter on asymptotics · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed. · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, 3rd ed. · Dingle, Asymptotic Expansions: Their Derivation and Interpretation (1973).

Papers. Boyd, J.P. (1999), "The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series", Acta Applicandae Mathematicae 56, 1 · Berry, M.V. (1989), "Uniform asymptotic smoothing of Stokes's discontinuities", Proc. R. Soc. Lond. A 422, 7 · Berry & Howls (1990), "Hyperasymptotics", Proc. R. Soc. Lond. A 430, 653.

Provenance. The least-term bound and the Euler/Stieltjes worked examples follow Bender & Orszag; the superasymptotic and hyperasymptotic terminology and scaling follow Boyd (1999) and Berry–Howls (1990); the erfc smoothing of the Stokes jump follows Berry (1989). A verifier should recompute: (i) the term-ratio condition $(n+1)/A|z|=1$ giving $N_\star\approx A|z|$ ; (ii) the Stirling evaluation $\varepsilon_{\min}\sim\sqrt{2\pi A|z|}\,e^{-A|z|}$ ; (iii) the Euler series at $z=10$ — least term $3.63\times10^{-4}$ , bracket $S_9=0.915456<f(10)=0.915633<S_{10}=0.915819$ , midpoint $0.915638$ ; (iv) Stirling's series at $z=10$ — $A=2\pi$ , $N_\star\approx31$ , floor $e^{-20\pi}\approx5\times10^{-28}$ .

Prerequisites. Asymptotic Series and Poincaré Expansions · Order Symbols and Gauge Functions · Watson's Lemma · Laplace's Method for Integrals. Leads to. Stokes Phenomenon and Anti-Stokes Lines · Borel Summation and Resurgence · WKB Approximation · Method of Steepest Descent and Saddle Points.

Ask the tutor: Ask me to walk through the least-term calculation for any asymptotic series you meet — the Airy function, a WKB expansion, or a perturbation series — and I will show you exactly where to stop and how small the error gets.

What is the difference between a convergent series and an optimally truncated asymptotic one?

A convergent series gets arbitrarily accurate if you add enough terms — the limit exists and equals the function. A divergent asymptotic series has no limit at fixed $z$ : its partial sums improve for a while, reach a best accuracy at the least term, then diverge. So a convergent series is truncated for convenience, while an asymptotic series must be truncated at the least term because more terms actively hurt. Paradoxically, an optimally truncated divergent series can beat a slowly convergent one with a handful of terms, which is why asymptotic expansions dominate large-argument computation.

Why is the error roughly the size of the first omitted term?

For an alternating series with decreasing terms, consecutive partial sums lie on opposite sides of the true value, so $f$ is trapped in a bracket whose width is exactly the next term. Truncating at order $N$ leaves an error no larger than $t_{N+1}$ , and because the true value usually sits near the middle of the bracket, the error is comparable to that term, not just bounded by it. This is what makes the least term a usable error estimate: you can read the accuracy of your answer straight off the size of the first term you dropped, without ever knowing $f$ . For non-alternating series the statement is heuristic but still the right order of magnitude.

Where does the factorial growth of the coefficients come from?

Factorial growth $a_n\sim n!/A^n$ is the generic signature of asymptotic series from saddle points and perturbation theory. In an integral like $\int e^{-t}(1+t/z)^{-1}\mathrm{d}t$ each term of the geometric expansion contributes a moment $\int_0^\infty e^{-t}t^n\mathrm{d}t=n!$ , so the factorial is inevitable. More deeply, the Borel transform has a singularity a finite distance $A$ from the origin, and a singularity at radius $A$ forces the coefficients to grow like $n!/A^n$ . In physics that singularity is an instanton, a renormalon, or a second saddle — so the divergence is information, not pathology.

What exactly do superasymptotics and hyperasymptotics mean?

Superasymptotics is optimal truncation of a single series: stop at the least term and you achieve an error exponentially small, $\sim e^{-A|z|}$ , beyond every algebraic order. That exponential is the floor for one series. Hyperasymptotics goes below it by taking the remainder seriously: it re-expands that remainder in a second divergent series multiplied by the subdominant exponential $e^{-Az}$ , truncates that optimally, and iterates, each stage descending a further slice of the exponent. Both terms are due to Berry and Howls; superasymptotics is the endpoint of ordinary asymptotics, hyperasymptotics the controlled path into exponential asymptotics.

Is optimal truncation a rule of thumb or something rigorous?

Both, depending on the series. For alternating series whose terms genuinely decrease then increase — Stieltjes series, the Euler series, many steepest-descent expansions — the least-term bound is a theorem: the remainder is provably no larger than the first omitted term, so truncating there is rigorously optimal up to a constant. For general series the argument that the error tracks the least term is heuristic, but backed by the Borel-transform picture, which locates the remainder precisely and confirms the $e^{-A|z|}$ scaling. In practice one truncates at the least term and, where rigour is needed, supplies an explicit remainder bound from an integral representation.

What does the exponentially small remainder physically mean?

It is almost never noise — it is a real, if faint, physical contribution that no term of the power series can see. In WKB it is the tunneling amplitude $e^{-S_0/\hbar}$ through a classically forbidden barrier. In an integral with two saddle points it is the subdominant saddle, $\sim e^{-(\Phi_2-\Phi_1)}$ , which the dominant saddle's expansion cannot contain. In quantum field theory it is an instanton or renormalon effect setting the nonperturbative scale. Because these effects switch on across Stokes lines, they are invisible to any finite order of perturbation theory yet decisive for the answer — which is why exponential asymptotics is a working tool, not a curiosity.