We derive the first four terms of an expansion in m_H²/M_t² of the total Higgs cross-section through gluon fusion. At NLO we demonstrate the excellent convergence of this series to the known result keeping the exact top mass dependence. At NNLO there is no known exact result, and our work represents a thorough quantitative investigation of the effects of finite top mass at this order. We discuss the applicability of our approach, and present numerical results for the LHC and Tevatron.

PACS numbers: 12.38.Bx, 14.80.Bn