Quantitative Integral Comparison for Monotone Series III: A Weak Euler–Maclaurin Decomposition via Convex Trapezoid Bounds

Jefferson, Bob (2026).
Quantitative Integral Comparison for Monotone Series III: A Weak Euler–Maclaurin Decomposition via Convex Trapezoid Bounds.

Abstract

This project develops a minimal Euler–Maclaurin type framework relating discrete sums to their integral approximations using only elementary analytic structure.

For functions that are decreasing on [1,N][1,N][1,N] and convex on each unit interval [k−1,k][k-1,k][k−1,k], the partial sums admit the decomposition

∑k=1Nf(k)=∫1Nf(x) dx+f(1)+f(N)2+E(N),\sum_{k=1}^{N} f(k) = \int_{1}^{N} f(x)\,dx + \frac{f(1)+f(N)}{2} + E(N),k=1∑Nf(k)=∫1Nf(x)dx+2f(1)+f(N)+E(N),

where the correction term satisfies the explicit bounds

0≤E(N)≤f(1)−f(N)2.0 \le E(N) \le \frac{f(1)-f(N)}{2}.0≤E(N)≤2f(1)−f(N).

This identity isolates a simple Euler–Maclaurin structure: the trapezoid endpoint correction appears naturally, while the remainder is controlled by a global bound requiring only convexity on unit intervals.

Method

The argument relies on three elementary analytic ingredients:

• decomposition of the integral into unit intervals
• a convex trapezoid estimate on each unit interval
• summation of these local bounds to obtain a global discrepancy estimate

Unlike the classical Euler–Maclaurin formula, the result avoids higher derivatives, Bernoulli numbers, and asymptotic expansions. The analysis uses only monotonicity, convexity, and basic properties of integrals.

The resulting decomposition provides a transparent structural relationship between discrete sums and integral approximations while remaining compatible with formal verification.

Examples

The framework yields explicit bounds for concrete monotone functions such as

f(x)=1x2,f(x)=1x3,f(x)=\frac{1}{x^{2}}, \qquad f(x)=\frac{1}{x^{3}},f(x)=x21,f(x)=x31,

illustrating how the decomposition controls the discrepancy between partial sums and their corresponding integral approximations.

Formal verification

All results are formally verified in Lean 4 using mathlib4.

The formal development proves:

• the decomposition of integrals into unit intervals
• the convex trapezoid estimate on each unit interval
• the weak Euler–Maclaurin decomposition
• explicit bounds for the correction term E(N)E(N)E(N)

The Lean library accompanying the paper provides machine-checked verification of the analytic arguments and exposes the main results as reusable theorems for integral comparison estimates.

Repository contents

The archive includes:

• Lean 4 source files implementing the formal proofs
• the accompanying research paper (LaTeX source and compiled PDF)
• documentation mapping Lean theorems to mathematical statements
• build instructions and project metadata

The development builds with Lean 4 and mathlib4.

Scope

This project lies at the intersection of

• classical analysis
• analytic number theory
• formal verification of mathematics

and illustrates how elementary analytic comparison principles can be formalised in a proof assistant while preserving a transparent mathematical structure.