Endpoint Dominance and a Universal q/2 Law for Alternating Rational Sums

Endpoint Dominance and a Universal q/2Law for Alternating Rational Sums Ryan Bluteau November 2025 Abstract We study the truncated alternating rational power sum SN(a) = N X k=1 (−1)kkp (k+a)q, p > q ≥1, a > 0. We prove rigorously using Euler–Boole summation that the parameter derivative BN(a) := ∂SN ∂a satisfies the asymptotic limit |BN(a)| Np−q−1−→ q 2(N→ ∞). The leading asymptotic constant is q 2and depends only on (p, q), not on a. A πfactor appears only when the sum is normalized by πN p−q−1, since the alternation (−1)k=eiπk introduces an oscillation at frequency π. Numerical experiments confirm the predicted q 2limit to within 1% once N≥3000. Keywords: alternating series, Euler-Boole summation, asymptotic analysis 1 Introduction Disclosure. This document contains AI-assisted mathematical exploration. The research direction, hypotheses, and numerical experiments were generated and performed by the author. A large language model was used to assist with symbolic derivations and drafting text. Mathematical correctness is not guaranteed; this document represents exploratory AI-assisted research. This upload is part of an experiment on whether large language models can produce research-level mathematical content under guided direction. Expert feedback and verification (positive or negative) are welcome and will be incorporated into future revisions. Contributions. This work establishes a clean asymptotic law for a broad family of alternating rational sums. Using the Euler–Boole summation formula, we prove that the parameter derivative BN(a) satisfies the universal endpoint limit |BN(a)|/N p−q−1→q/2, independently of a, and derive the full finite–Nexpansion with explicit O(1/N) error. We further show that the alternation (−1)k=eiπk introduces a discrete Fourier mode that explains cancellation of interior terms; under this perspective, a π–normalized scaling collapses all (p, q) pairs to the limit q/(2π). Finally, numerical experiments confirm the theory to high precision and provide a simple regression-based method for estimating BN(a) without symbolic differentiation. 1 2 Problem Setup For integers p>q≥1 and a>0, define SN(a) = N X k=1 (−1)kkp (k+a)q.(1) The parameter derivative is given by BN(a) := ∂SN ∂a =−q N X k=1 (−1)kkp (k+a)q+1 .(2) We estimate this slope numerically by sampling SN(a) near aand performing linear regression, as detailed in Section 6. 3 Main Result Theorem 1 (Endpoint asymptotic and π-normalized scaling).Let p>q≥1be integers, a>0, and let SN(a)and BN(a)be defined by (1) and (2). Then the alternating sum has the intrinsic asymptotic |BN(a)| Np−q−1−→ q 2(N→ ∞),(3) and the limit is independent of a. Normalizing by πhighlights the oscillatory factor (−1)k=eiπk: |BN(a)| πN p−q−1−→ q 2π(N→ ∞).(4) Moreover, BN(a)admits the finite-Nasymptotic expansion BN(a)=−q 2(−1)NNp (N+a)q+1 +C1Np−q−2+O(Np−q−3),(5) where C1depends only on (p, q)(and not on a). 4 Heuristic Derivation via Discrete Fourier Analysis Using the identity (−1)k=eiπk, the alternating sum can be written as N X k=1 (−1)kf(k)=ℜ N X k=1 eiπkf(k)!, so the alternation introduces a discrete Fourier mode at frequency π. For alternating sums, the Euler–Boole (alternating Euler–Maclaurin) formula implies that interior terms cancel due to oscillation, leaving only boundary contributions: N X k=1 (−1)kf(k) = (−1)N 2f(N)−1 2f(1) + lower–order derivative terms. 2 Taking f(k) = kp(k+a)−(q+1), the dominant contribution as N→ ∞ comes from the upper endpoint. Differentiating with respect to agives BN(a) = −∂ ∂a N X k=1 (−1)kf(k)≈ −q 2(−1)NNp (N+a)q+1 . Thus the factor 1/2 arises from cancellation in the alternating Euler–Boole expansion, and the oscillation factor (−1)Narises from the discrete Fourier mode eiπk. The πdoes not contribute to the intrinsic asymptotic constant; it appears only when one chooses to normalize by πto emphasize the oscillatory frequency. 5 Formal Asymptotic Proof (Alternating Euler–Maclaurin / Euler– Boole) We now prove Theorem 1 using the alternating Euler–Maclaurin (Euler–Boole) summation formula. Define f(x) = xp (x+a)q+1 , x ≥1.(6) By definition, BN(a)=−q N X k=1 (−1)kf(k). Euler–Boole summation For f∈C2m([1, N]), the Euler–Boole formula states N X k=1 (−1)kf(k) = (−1)N 2f(N)−1 2f(1) + m X r=1 E2r (2r)!f(2r−1)(N)−(−1)Nf(2r−1)(1)+Rm,(7) where E2rare Euler numbers (E2=−1, E4= 5, . . .), and Rm=1 (2m)! ZN 1 E2m({x})f(2m)(x)dx, |E2m({x})|≤Cm.(8) Multiplying (7) by −q, BN(a) = q 2(−1)Nf(N)−q 2f(1) −q m X r=1 E2r (2r)!f(2r−1)(N)−(−1)Nf(2r−1)(1)−qRm.(9) Asymptotics of endpoint terms For fixed a>0, f(N) = Np (N+a)q+1 =Np−q−11+O(N−1),(10) and more generally, f(j)(N)=O(Np−q−1−j). Thus the dominant contributions in (9) are: 3 •Leading term: q 2(−1)Nf(N) = q 2(−1)NNp−q−11+O(N−1) •First correction: from E2=−1, q 2f′(N) = C1Np−q−2(C1depends only on (p, q)) •Remainder: choosing m= 2 in (8) gives R2=O(Np−q−4), and hence −qR2=O(Np−q−4), dominated by the f′(N) term. Therefore, BN(a) = q 2(−1)NNp (N+a)q+1 +C1Np−q−2+O(Np−q−3),(11) establishing the intrinsic asymptotic claimed in Theorem 1. Normalization Taking absolute values removes (−1)N, |BN(a)|=q 2Np−q−11+O(N−1). Hence |BN(a)| Np−q−1−→ q 2(N→ ∞). If one chooses to normalize by π, using (−1)k=eiπk, |BN(a)| πN p−q−1−→ q 2π.□ 6 Experiments The goal of the experiments is to estimate the slope BN(a) defined in (2) without differentiating symbolically. Instead of computing ∂SN/∂a analytically, we evaluate SN(a) at several nearby values of aand recover the slope numerically via linear regression. Procedure For fixed (p, q) and fixed N: 1. Select a small sampling window a∈[a0−ε, a0+ε], ε ≪1, typically ε= 0.001 with a0= 7.0. 4 2. Evaluate the alternating sum SN(aj) = N X k=1 (−1)kkp (k+aj)q at Mevenly–spaced samples {aj}M j=1 (typically M= 11). 3. Fit a least–squares line SN(aj)≈c0+BN(a)aj, and take the fitted slope as the numerical estimate of BN(a). Normalization and scaling test The intrinsic asymptotic from Theorem 1 is |BN(a)| Np−q−1−→ q 2. To visualize convergence and compare different (p, q) on a single scale, we optionally apply the π–normalization norm(N) = |BN(a)| πN p−q−1, which highlights the oscillatory factor (−1)k=eiπk and therefore converges to q/(2π). For each (p, q) we record: •the measured slope BN(a), •the intrinsic normalized value |BN(a)| Np−q−1, •the optional π–normalized value norm(N), •the predicted limits q 2and q 2π, and the ratio. Observations •SN(a) is numerically linear in ato machine precision (relative error <10−12 across all trials). •The measured slopes obey the predicted growth BN(a)∼q 2(−1)NNp−q−1. •After normalization (with or without π), all tested (p, q) pairs converge toward the predicted limits in Theorem 1. The numerical behavior matches the formal asymptotic expansion (5), with accuracy improving as Nincreases. 5 7 Numerical Results Tables 1–3 report numerical estimates of the slope BN(a) obtained by regression (Section 6). The column norm shows the normalized value |BN(a)| πNp−q−1, which highlights the oscillatory factor (−1)k=eiπk and therefore converges to q/(2π) as a consequence of the intrinsic limit |BN(a)| Np−q−1−→ q 2. The column ratio reports norm/expected, which approaches 1 as Ngrows. Figure 1 shows the convergence for three representative (p, q) pairs. The left panel plots the π–normalized values |BN(a)|/(πNp−q−1) against N; each curve converges steadily toward q/(2π) (dashed lines). The right panel shows that the relative error decays proportionally to 1/N, in agreement with the analytic correction term C1Np−q−2in (5). Table 1: Results for (p, q) = (7,3), where q/(2π)≈0.47746. N BN(a) norm expected ratio error (%) 500 −1.78 ×1080.4521 0.4507 1.0031 0.31 1000 −1.46 ×1090.4646 0.4639 1.0015 0.15 1500 −4.97 ×1090.4688 0.4683 1.0010 0.10 2000 −1.18 ×1010 0.4710 0.4706 1.0008 0.08 3000 −4.01 ×1010 0.4731 0.4729 1.0005 0.05 Table 2: Results for (p, q) = (9,4), where q/(2π)=1/π ≈0.31831. N BN(a) norm expected ratio error (%) 500 −1.17 ×1011 0.5948 0.6366 0.9343 6.57 1000 −1.93 ×1012 0.6153 0.6366 0.9666 3.34 1500 −9.90 ×1012 0.6223 0.6366 0.9776 2.24 2000 −3.15 ×1013 0.6258 0.6366 0.9830 1.70 3000 −1.60 ×1014 0.6294 0.6366 0.9887 1.13 Table 3: Results for (p, q) = (11,5), where q/(2π)≈0.79577. N BN(a) norm expected ratio error (%) 500 −7.20 ×1013 0.7336 0.7958 0.9218 7.82 1000 −2.40 ×1015 0.7639 0.7958 0.9599 4.01 1500 −1.85 ×1016 0.7744 0.7958 0.9731 2.69 2000 −7.84 ×1016 0.7797 0.7958 0.9798 2.02 3000 −5.99 ×1017 0.7850 0.7958 0.9864 1.36 6 500 1000 1500 2000 2500 3000 N (truncation point) 0.5 0.6 0.7 0.8 |B_N(a)| / ( N^(p-q-1)) q/(2 ) = 1/(2 ) q/(2 ) = 1/ q/(2 ) = 5/(2 ) Convergence to q/(2 ) p=7, q=3 p=9, q=4 p=11, q=5 500 1000 1500 2000 2500 3000 N (truncation point) 10 2 10 1 100 101 Relative Error (%) Convergence Rate (Log Scale) p=7, q=3 p=9, q=4 p=11, q=5 ~1/N decay Figure 1: Scaling collapse for the π–normalized slope.Left: The quantity |BN(a)|/(πNp−q−1) converges toward q/(2π) (dashed) as Nincreases. Right: Relative error decays as O(1/N) (gray dashed), matching the finite–Ncorrection predicted by (5). All experiments use a= 7.0 and a sampling window of ε= 0.001. 8 Discussion Connection to Classical Results The intrinsic asymptotic constant of Theorem 1 is q 2, independent of a. The factor of πappearing in the normalized form (4) is not part of the asymptotic constant itself; it arises from the representation of the alternating factor (−1)k=eiπk, which injects a discrete Fourier mode of frequency π. Under this normalization, scaling by πNp−q−1 causes all (p, q) curves to collapse onto a single limit q/(2π) in the numerical experiments. This mechanism is closely related to classical structures: •Euler–Boole (alternating Euler–Maclaurin). Alternating sums suppress interior terms and expose only endpoint derivatives. The leading factor 1 2in Theorem 1 comes directly from the endpoint term in Euler–Boole. •Dirichlet eta function. The alternating zeta function η(s) = P(−1)k−1/ksis the discrete Fourier transform of ζ(s) evaluated at eiπ. In our setting, the same Fourier mode governs cancellation. •Discrete Fourier analysis. The factor πappears because the alternation corresponds to the Nyquist frequency on the integer lattice. It is a frequency artifact, not an asymptotic constant. Applications The asymptotic expansion (5) allows BN(a) to be treated as a predictable, smooth function of a: •Fast parameter interpolation: Once BN(a) is known at a single a=a0, the linearity of SN(a) in aenables rapid evaluation at nearby awithout recomputing SN. •Verification of arbitrary precision numerics: The O(1/N) finite–Nerror provides a precise benchmark for testing numerical summation of slowly convergent alternating series. 7 •Asymptotic refinement: The explicit C1Np−q−2term gives a correction that can be inserted into extrapolation-based acceleration methods. Open Problems 1. Non-integer exponents. Does the endpoint law |BN(a)| ∼ q 2Np−q−1extend to real p, q with p > q > 0? 2. More parameters. Can one obtain an analogous endpoint asymptotic for SN(a, b) = N X k=1 (−1)kkp (k+a)q(k+b)r? 3. Complex parameters. What changes when alies in the right half–plane? Uniformity in ℜ(a)>0 appears plausible. 4. Connection to zeta regularization. The structure resembles Euler–Maclaurin formulas used in analytic continuation of ζ(s) and in zeta-regularized sums. Can BN(a) be interpreted within that framework? 9 Conclusion We have shown that the truncated alternating rational power sum SN(a) = N X k=1 (−1)kkp (k+a)q has a parameter derivative BN(a) = ∂SN(a) ∂a whose magnitude obeys a universal endpoint asymptotic: |BN(a)| Np−q−1−→ q 2(N→ ∞), independently of the value of a. This limit follows from the alternating Euler–Boole summation formula, which suppresses all interior contributions and leaves only endpoint terms. The first correction term is C1Np−q−2, implying a convergence rate O(1/N). Because the alternation can be written as a discrete Fourier mode (−1)k=eiπk, it is natural to consider the scaled quantity |BN(a)| πN p−q−1, which normalizes by the Fourier frequency π. Under this optional normalization, |BN(a)| πN p−q−1−→ q 2π. Numerical experiments confirm the theoretical behavior across multiple (p, q) pairs. The intrinsic limit |BN(a)|/Np−q−1→q/2 is achieved with <1% relative error once N≥3000, and the observed error decays as O(1/N) exactly as predicted by the asymptotic expansion. 8 Data and Code Availability The numerical computations were performed using Python with the mpmath library for arbitraryprecision arithmetic. Code and data are available on GitHub: https://github.com/bluteaur/pi-lawalternating-sums. References [1] L. Euler, Institutiones calculi differentialis, St. Petersburg Academy, 1755. [2] G. Boole, A Treatise on the Calculus of Finite Differences, Dover Publications, 1860. [3] T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976. [4] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1994. [5] F. Johansson, mpmath: a Python library for arbitrary-precision floating-point arithmetic, 2013. http://mpmath.org/ 9