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The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic functions in another. Similar results are also given for the midpoint rule.

A trapezoidal rule error bound unifying the Euler–Maclaurin formula and geometric convergence for periodic functions