In this paper,the initial layer problem and infinite Prandtl number limit of Rayleigh-Bénard convection is studied by the asymptotic expansion methods of singular perturbation theory and the classical energy methods.For ill-prepared initial data,an exact approximating solution with expansions up to any order are given and the convergence rates O(εm+12)and the optimal convergence rates O(εm+1) are obtained respectively.This improves the result of J.G.SHI.
