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. 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 JGSHI.