  We derive the first four terms of an expansion in mH^2/mtop^2 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.
