An Exact Trapezoidal-Correction Series for Definite Integration: A Convergence Theorem and Explicit Error Bounds

This mathematical note presents an exact representation of definite integrals through an infinite series of successive trapezoidal corrections, providing an antiderivative-free approach to evaluating continuous functions. For a continuous function \(f:[a,b]\to\mathbb{R}\), define \(h_n=(b-a)/2^n\) and \(x_{n,k}=a+kh_n\). The principal identity is \(\int_a^b f(x)\,dx=\frac{b-a}{2}[f(a)+f(b)]+\sum_{n=0}^{\infty}D_n(f)\), where \(D_n(f)=\frac{h_n}{4}\sum_{k=0}^{2^n-1}\left[2f\left(x_{n,k}+\frac{h_n}{2}\right)-f(x_{n,k})-f(x_{n,k}+h_n)\right]\). The note establishes convergence for every continuous function on a compact interval by expressing the correction series as a telescoping sequence of composite trapezoidal approximations. For twice continuously differentiable functions satisfying \(|f''(x)|\le M\), an explicit truncation error bound is proved: \(\left|\int_a^b f(x)\,dx-T_N\right|\le\frac{M(b-a)^3}{12\cdot4^N}\). Applications to \(f(x)=x^2\) and \(f(x)=e^{-x^2}\) illustrate exact evaluation and convergent approximation without requiring elementary antiderivatives. The representation is mathematically equivalent to classical trapezoidal refinement and is presented as an expository formulation rather than a claim of a new integration theorem. Its purpose is to provide a self-contained derivation, rigorous convergence proof, and quantitative error estimates for an alternative computational representation of definite integration.