Asymptotic Series & Optimal Truncation

Derivation

Asymptotic Series & Optimal Truncation

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Statement

A function \(f(x)\) has a Poincaré asymptotic expansion \(f(x) \sim \sum_{n=0}^{\infty} a_n x^{-n}\) as \(x \to \infty\) if, for every fixed order \(N\), the remainder after \(N+1\) terms is \(o(x^{-N})\), i.e. \(\left| f(x) - \sum_{n=0}^{N} a_n x^{-n}\right| = o(x^{-N})\). Such a series may diverge for every finite \(x\), yet its partial sums approximate \(f\) with an error controlled by the first omitted term. We derive this and show that truncating at the least term minimises the error, giving the optimal-truncation bound \(\varepsilon_{\text{opt}} \approx |a_{N^\*} x^{-N^\*}|\) at the order \(N^\*\) where the terms stop decreasing.

Why it matters

Most perturbation series in physics — the Stirling expansion of \(\ln\Gamma\), the WKB series, the high-temperature or high-field expansions of statistical mechanics, and the perturbation series of quantum electrodynamics — are divergent. Naively they seem worthless. The Poincaré framework rescues them: read as asymptotic series they deliver spectacular accuracy from a handful of terms, provided one stops at the right place.

The practical payload is the least-term rule. It tells you not to keep adding terms of a divergent series but to truncate where they are smallest, and it hands you an honest error bar equal to that smallest term. This is the single most useful fact about divergent series in physics.

Assumptions

The remainder is bounded by the first omitted term.If \(|R_N(x)| \le C\,|a_{N+1} x^{-(N+1)}|\) fails, the least-term estimate of the error loses its meaning and truncation gives no controlled bound. The coefficients \(a_n\) are uniquely fixed by \(f\).The asymptotic series is unique, but the function is not: dropping this leaves \(f\) and \(f + e^{-x}\) sharing the same series, so the expansion cannot resolve exponentially small ("beyond all orders") content. The coefficients eventually grow, typically factorially, \(a_n \sim n!\,A^{-n}\).If they did not grow the series would converge and there would be no optimal truncation; factorial growth is what creates a genuine minimum term at finite \(N^\*\). \(x\) is large enough that a minimum term exists in the accessible range.If \(x\) is too small the terms only increase and no optimal truncation order is available; the expansion is then useless at that argument.

Derivation

\[ f(x) = \sum_{n=0}^{N} a_n x^{-n} + R_N(x), \qquad R_N(x) \equiv f(x) - \sum_{n=0}^{N} a_n x^{-n}. \]

Definition: split \(f\) into its order-\(N\) partial sum \(S_N\) and an exact remainder \(R_N\). No approximation yet. A

\[ \text{Poincaré condition:} \quad \lim_{x\to\infty} x^{N} R_N(x) = 0 \quad \text{for each fixed } N. \]

This is the definition of \(f \sim \sum a_n x^{-n}\): for fixed \(N\), the remainder vanishes faster than the last kept term as \(x\to\infty\). Note the order of limits — \(N\) fixed, then \(x\to\infty\). A

\[ a_N = \lim_{x\to\infty} x^{N}\left( f(x) - \sum_{n=0}^{N-1} a_n x^{-n} \right). \]

Solve the Poincaré condition recursively for the coefficients: subtract the lower-order sum, multiply by \(x^{N}\), take the limit. This shows the \(a_n\) are uniquely and successively determined by \(f\). B

\[ R_N(x) = a_{N+1} x^{-(N+1)} + R_{N+1}(x), \qquad R_{N+1}(x) = o\!\left(x^{-(N+1)}\right). \]

Peel one more term off the remainder using step 1 at order \(N+1\). For large \(x\) the leading piece is the first omitted term \(a_{N+1}x^{-(N+1)}\). B

\[ |R_N(x)| \le |a_{N+1}|\,x^{-(N+1)} \bigl(1 + o(1)\bigr) \;\approx\; |u_{N+1}(x)|, \qquad u_n(x)\equiv a_n x^{-n}. \]

For many physical series (alternating with monotone terms, or Borel-summable factorial series) the remainder is bounded in magnitude by the first omitted term \(u_{N+1}\); for strictly alternating monotone series this is the exact Leibniz bound. Denote the \(n\)th term \(u_n\). C

\[ \left|\frac{u_{n+1}}{u_n}\right| = \frac{|a_{n+1}|}{|a_n|}\,x^{-1} \sim \frac{n+1}{|A|\,x} \quad\text{for}\quad a_n \sim n!\,A^{-n}. \]

Form the ratio of successive terms and insert factorial growth. The terms shrink while this ratio is below one and grow once it exceeds one. C

\[ \left|\frac{u_{N^\*+1}}{u_{N^\*}}\right| = 1 \quad\Longrightarrow\quad \frac{N^\*+1}{|A|\,x} \approx 1 \quad\Longrightarrow\quad \boxed{\,N^\* \approx |A|\,x\,}. \]

The optimal truncation order \(N^\*\) is where the terms stop decreasing (the least term). Set the ratio to one and solve. The best partial sum keeps roughly \(|A|x\) terms. C

\[ \varepsilon_{\text{opt}} \approx |u_{N^\*}| = |a_{N^\*}|\,x^{-N^\*} \sim N^\*!\,|A|^{-N^\*} x^{-N^\*}. \]

The optimal error equals the least term. Substitute \(a_{N^\*}\sim N^\*!\,A^{-N^\*}\) from the assumed factorial growth. C

\[ N^\*!\,|A|^{-N^\*}x^{-N^\*} \;\approx\; \sqrt{2\pi N^\*}\,\left(\frac{N^\*}{e\,|A|\,x}\right)^{N^\*} \;\approx\; \sqrt{2\pi |A| x}\;\, e^{-|A|x}. \]

Insert Stirling's formula \(N^\*!\approx\sqrt{2\pi N^\*}(N^\*/e)^{N^\*}\) and \(N^\*\approx |A|x\). The powers of \(x\) and \((N^\*/e)^{N^\*}\) collapse, leaving an exponentially small residual. C

\[ \boxed{\;\varepsilon_{\text{opt}}(x) \;\sim\; \sqrt{2\pi |A| x}\;\, e^{-|A|\,x}\;} \qquad (x\to\infty). \]

Optimal truncation of a factorially divergent series achieves an error that is exponentially small in \(x\) — beyond every power of \(1/x\). This is the fundamental optimal-truncation error bound. C

Result

\[ |f(x)-S_N(x)| \lesssim |a_{N+1}|\,x^{-(N+1)}, \quad N^\* \approx |A|\,x, \quad \varepsilon_{\text{opt}} \sim \sqrt{2\pi|A|x}\; e^{-|A|x}. \]

Reading. Stop a divergent asymptotic series at its smallest term, near order \(N^\*\approx|A|x\). The error incurred is about the size of that smallest term. For factorial coefficients this least-term error is exponentially small in \(x\), so a divergent series gives super-algebraic accuracy — but no better, because the exponentially small piece \(e^{-|A|x}\) is invisible to the power series and sets a hard floor.

Units check. The argument \(x\) is dimensionless (it is the large expansion parameter, e.g. \(z\) in Stirling or \(1/\hbar\)-type ratios), each term \(a_n x^{-n}\) carries the dimensions of \(f\), and \(N^\*=|A|x\) is a pure number as an order-count must be. The bound \(\varepsilon_{\text{opt}}\) shares the dimensions of \(f\); \(|A|x\) inside the exponential is dimensionless, as required.

Limiting cases

Convergent limit \(a_n\) bounded (no factorial growth): the ratio \(|u_{n+1}/u_n|\to 0/x<1\) for all \(n\), so \(N^\*\to\infty\), there is no least term, and the series converges to \(f\) exactly — the ordinary Taylor picture.
\(x\to\infty\) at fixed \(N\): \(R_N\sim a_{N+1}x^{-(N+1)}\to 0\); the Poincaré definition is recovered and any fixed truncation is asymptotically exact.
\(x\to\infty\) with optimal \(N=N^\*(x)\): error \(\sim e^{-|A|x}\to 0\) exponentially — the best a divergent series can do.
Alternating monotone terms: the Leibniz bound makes step 5 exact, \(|R_N|\le|u_{N+1}|\), with the true value bracketed between consecutive partial sums.
Small \(x\lesssim 1/|A|\): \(N^\*\lesssim 1\); only the first term or two is usable and the expansion carries little information.

Breaks when

Terms never decrease. If \(x\) is too small (\(x\lesssim 1/|A|\)) the ratio \(|u_{n+1}/u_n|>1\) from the start; there is no minimum term, no optimal truncation, and the partial sums diverge monotonically away from \(f\). The bound is empty.
Exponentially small structure is the answer. When the physics lives in the \(e^{-|A|x}\) term itself — Stokes phenomena, instanton contributions, spectral splittings, tunnelling rates — the asymptotic series is blind to it. Optimal truncation bounds the ambiguity but cannot compute the effect; one needs Borel summation, resurgence, or exponential asymptotics.
Coefficients grow faster than factorially (e.g. \(a_n\sim (2n)!\) or \(a_n\sim (n!)^2\)). Then \(N^\*\) is pushed to very low order and \(\varepsilon_{\text{opt}}\) is not small; the series is not Borel-summable in the ordinary sense and the least-term error can remain \(O(1)\).
The remainder is not controlled by the first omitted term. If the series is neither alternating-monotone nor Borel-summable, the constant \(C\) in \(|R_N|\le C|u_{N+1}|\) can be large or unbounded, and truncating at the least term no longer bounds the actual error.

Failure modes

"Add more terms for more accuracy." Summing past \(N^\*\) makes a divergent series worse; accuracy improves only up to the least term, then degrades factorially.
Confusing radius of convergence with usefulness. A zero radius of convergence (divergent everywhere) does not mean the series is useless — it is asymptotic and highly accurate when truncated correctly.
Swapping the limit order. Poincaré requires \(N\) fixed then \(x\to\infty\); students who take \(N\to\infty\) at fixed \(x\) "prove" divergence and wrongly discard the series.
Forgetting the exponential floor. Reporting an error smaller than \(e^{-|A|x}\) from a power series alone; no truncation can beat the beyond-all-orders term.
Locating \(N^\*\) by the term value instead of the ratio. Using \(|u_n|\) minimisation is fine, but confusing it with "where terms are small" rather than "where the ratio crosses one" misplaces \(N^\*\) for slowly varying \(x\).
Assuming uniqueness of \(f\). Treating the recovered series as if it determined \(f\) exactly, ignoring that \(f+ce^{-x}\) has the identical expansion.

Discussion

The deep lesson is that convergence and usefulness are independent properties of a series. A Taylor series (the prior result, taylor-series-analytic) converges inside a disc set by the nearest singularity; an asymptotic series need not converge anywhere yet can approximate a function to exponential accuracy in a sector. The Poincaré definition trades "sum of infinitely many terms equals \(f\)" for "finite truncation error is uniformly small in the large parameter" — a weaker but far more common and physically relevant guarantee.

Factorial growth of coefficients, \(a_n\sim n!\,A^{-n}\), is not an accident. It arises whenever the coefficients are moments of a function with a singularity, \(a_n=\int_0^\infty t^n \rho(t)\,dt\)-like structures, and \(A\) marks the location of the nearest singularity of the Borel transform \(B(t)=\sum a_n t^n/n!\), which has finite radius \(|A|\). The optimal error \(e^{-|A|x}\) is precisely the exponential set by that Borel singularity: the least-term barrier and the Borel-plane singularity are two faces of the same object. This is the entry point to resurgence theory.

The exponentially small residual \(e^{-|A|x}\) is the physics that the perturbation series cannot see. In QED it is the scale of non-perturbative effects; in WKB it is the tunnelling amplitude through a classically forbidden region; in the Stokes phenomenon it is the term that switches on across an anti-Stokes line. Optimal truncation is the sharpest statement one can make with the power series alone, and its error bound \(\sqrt{2\pi|A|x}\,e^{-|A|x}\) is exactly the amplitude of the leading non-perturbative contribution — so the "error" of perturbation theory is the next layer of physics, recovered systematically by Borel–Écalle resummation.

Common misconceptions. Divergence does not mean the numbers are meaningless; it means you must not sum forever. The series does not "converge slowly" — it converges to nothing and only its partial sums are meaningful. And the least term is not merely a convenient stopping point: its magnitude is the intrinsic ambiguity of the asymptotic series, the price of representing \(f\) by powers of \(1/x\) alone.

Worked examples

\[ E_1(x)=\int_x^\infty \frac{e^{-t}}{t}\,dt = e^{-x}\sum_{n=0}^{\infty}\frac{(-1)^n n!}{x^{n+1}}, \qquad u_n=\frac{(-1)^n n!}{x^{n+1}}. \]

The exponential integral. Repeated integration by parts generates this divergent alternating series with \(a_n=(-1)^n n!\), so \(|A|=1\). Evaluate at \(x=5\). B

\[ N^\* \approx |A|\,x = 1\cdot 5 = 5, \qquad \left|\frac{u_{n+1}}{u_n}\right|=\frac{n+1}{x}=\frac{n+1}{5}=1 \Rightarrow n=4\text{–}5. \]

Least term near \(n\approx N^\*=5\); terms \(|u_n|=n!/5^{n+1}\) are smallest around \(n=4,5\) where they equal \(24/5^5=0.00768\) and \(120/5^6=0.00768\). B

\[ e^{x}E_1(5)\approx\sum_{n=0}^{4}\frac{(-1)^n n!}{5^{n+1}} = 0.2 - 0.04 + 0.016 - 0.0096 + 0.00768 = 0.17408. \]

Optimal partial sum truncated just before the least term. Multiply by \(e^{-5}\): \(E_1(5)\approx 0.17408\,e^{-5}=1.1727\times10^{-3}\). B

\[ E_1(5)\approx 1.173\times10^{-3}, \quad \varepsilon_{\text{opt}}\approx |u_5|e^{-5}=0.00768\,e^{-5}\approx 5.2\times10^{-5}. \]

Reading. The true value is \(E_1(5)=1.14830\times10^{-3}\); the optimal-truncation estimate errs by \(2.4\times10^{-5}\), comfortably inside the least-term bound \(5.2\times10^{-5}\). The least term itself, \(|u_5|=0.00768\), scaled by the \(e^{-5}\) prefactor gives \(5.2\times10^{-5}\) — the honest error bar delivered by the theory.

Units check. \(x=5\) dimensionless; \(E_1\) dimensionless; every term \(n!/x^{n+1}\) dimensionless. Consistent.

\[ \ln\Gamma(z)\sim\left(z-\tfrac12\right)\ln z - z + \tfrac12\ln 2\pi + \sum_{n=1}^{\infty}\frac{B_{2n}}{2n(2n-1)\,z^{2n-1}}, \]

The Stirling series, with Bernoulli numbers \(B_{2n}\). The correction terms grow factorially: \(B_{2n}\sim(-1)^{n+1}\,2(2n)!/(2\pi)^{2n}\), so effectively \(|A|=2\pi\) in the variable \(2\pi z\). Take \(z=10\). C

\[ c_n=\frac{B_{2n}}{2n(2n-1)z^{2n-1}}: \quad c_1=\frac{1}{12z}, \; c_2=-\frac{1}{360 z^3}, \; c_3=\frac{1}{1260 z^5},\dots \]

List the correction terms. At \(z=10\): \(c_1=8.333\times10^{-3}\), \(c_2=-2.778\times10^{-6}\), \(c_3=7.94\times10^{-9}\), \(c_4=-6.4\times10^{-11}\). C

\[ N^\*\approx 2\pi z = 2\pi\cdot 10 \approx 63 \;\text{(in the } 2n\text{ index, } n^\*\approx \pi z\approx 31). \]

The least term sits at extremely high order for \(z=10\) — the terms keep shrinking for a long time, so only round-off limits accuracy. Truncating after \(c_1\) already gives near-machine precision. C

\[ \ln\Gamma(10)\approx 9.5\ln 10 - 10 + \tfrac12\ln2\pi + \tfrac{1}{120} = 12.801827, \quad \varepsilon_{\text{opt}}\sim e^{-2\pi\cdot10}\approx 10^{-27}. \]

Reading. Exact \(\ln\Gamma(10)=\ln 9!=\ln 362880=12.8018275\). Keeping just the \(1/12z\) term already matches to seven digits; because \(N^\*\approx63\) is enormous, the exponential floor \(e^{-2\pi z}\sim10^{-27}\) is far below double precision — the Stirling series is superb for moderate \(z\).

Units check. \(z=10\) dimensionless; \(\ln\Gamma\) dimensionless; \(B_{2n}/[2n(2n-1)z^{2n-1}]\) dimensionless. Consistent.

Problems

(A) For the series \(\sum(-1)^n n!/x^{n+1}\) at \(x=3\), find the optimal truncation order \(N^\*\) and the least-term error bound.
Solution
\(N^\*\approx|A|x=1\cdot3=3\). The ratio \(|u_{n+1}/u_n|=(n+1)/3=1\) at \(n=2\text{–}3\). Least term \(|u_3|=3!/3^4=6/81=0.0741\) (or \(|u_2|=2/27=0.0741\), equal). So truncate near \(n=2\), \(\varepsilon_{\text{opt}}\approx 0.074\) (times \(e^{-3}\) if the \(e^{-x}\) prefactor applies, giving \(3.7\times10^{-3}\)). The small \(x\) means poor accuracy — only about one significant figure.
(B) A perturbation series has coefficients \(a_n=n!\,(1/2)^{-n}=n!\,2^{n}\). At what \(x\) does the optimal truncation keep exactly \(N^\*=10\) terms, and what is the exponential error floor there?
Solution
Here \(A^{-1}=2\Rightarrow A=1/2\), so \(|A|=1/2\). \(N^\*=|A|x\Rightarrow 10=x/2\Rightarrow x=20\). Error floor \(\varepsilon_{\text{opt}}\sim\sqrt{2\pi|A|x}\,e^{-|A|x}=\sqrt{2\pi\cdot10}\,e^{-10}=\sqrt{62.83}\,\cdot4.54\times10^{-5}=7.93\times4.54\times10^{-5}=3.6\times10^{-4}\).
(C) Show that if \(a_n=1/n^2\) (polynomially bounded), no optimal truncation order exists, and identify what kind of series this is.
Solution
The term ratio is \(|u_{n+1}/u_n|=\dfrac{n^2}{(n+1)^2}\,x^{-1}=\left(\dfrac{n}{n+1}\right)^2 x^{-1}<x^{-1}\le 1\) for all \(n\) when \(x\ge1\). The ratio never crosses one from below, so terms decrease monotonically forever: no least term, \(N^\*\to\infty\). The series \(\sum x^{-n}/n^2\) converges (radius of convergence \(1\); at \(x=1\) it sums to \(\pi^2/6\)). It is an ordinary convergent power series, not a genuinely asymptotic-divergent one — the optimal-truncation machinery is unnecessary.
(B) Estimate \(E_1(10)\) by optimal truncation of \(e^{x}E_1(x)=\sum(-1)^n n!/x^{n+1}\), and compare with the exact \(E_1(10)=4.15697\times10^{-6}\).
Solution
\(N^\*\approx x=10\). Terms \(|u_n|=n!/10^{n+1}\): \(u_0=0.1,\,u_1=0.01,\,u_2=0.002,\dots\), minimum near \(n=9,10\) where \(|u_9|=9!/10^{10}=3.6\times10^{-5}\). Summing to \(n=9\): \(e^xE_1(10)\approx 0.1-0.01+0.002-0.0006+0.00024-\dots\approx 0.09156\). Then \(E_1(10)\approx0.09156\,e^{-10}=0.09156\times4.54\times10^{-5}=4.157\times10^{-6}\), matching the exact value to four significant figures. Error bound \(\sim|u_9|e^{-10}=3.6\times10^{-5}\times4.54\times10^{-5}\approx1.6\times10^{-9}\).
(C) The coefficients of a series grow as \(a_n\sim(2n)!\). Explain why optimal truncation fails to give a small error, and name the resummation this signals.
Solution
With \(a_n\sim(2n)!\) the terms are \(|u_n|=(2n)!\,x^{-n}\). Ratio \(|u_{n+1}/u_n|=(2n+2)(2n+1)/x\), which equals one at \(N^\*\) where \((2N^\*)^2\approx x\), i.e. \(N^\*\approx\sqrt{x}/2\) — far lower order than the \(n!\) case, and the least term \(|u_{N^\*}|=(2N^\*)!\,x^{-N^\*}\) is not exponentially small; using Stirling it scales like \(e^{-c\sqrt{x}}\) at best, not \(e^{-|A|x}\). This is the signature of a series that is not Borel-summable along the standard \((n!)^{-1}\) transform: one needs a higher-order Borel transform (dividing by \((2n)!\), a Gevrey-2 series) or Écalle's resurgence with a generalised Borel plane. The optimal-truncation error remains large because the singularity in the ordinary Borel plane has collapsed to the origin.