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High-order scattering is found to be essential for the convergence of the multiple-scattering (MS) theory of x-ray-absorption fine structure, both in the near-edge and the extended regimes. These contributions are calculated using an ab initio curved-wave scattering-matrix formalism. Convergence to full MS accuracy is demonstrated for fcc Cu, as well as for molecular O 2 and N 2, where our approach provides a high-order MS interpretation of the σ* shape resonances.