In　this　paper　the　Initial　layer　problem　and　infinite　Prandtl　number　limit　of　Rayleigh-Bénard　convection　are　studied.　For　the　case　of　ill-prepared　initial　data　infinite　Prandtl　number　limit　of　the　Boussinesq　approximation　for　Rayleigh-Bénard　convection　is　proven　by　using　the　asymptotic　expansion　methods　of　singular　perturbation　theory　and　the　classical　energy　methods.　An　exact　approximating　solution　with　the　zero　order　term　and　the　1st　order　term　expansion　is　given　and　the　convergence　rates　O(ε3/2)　and　O(ε2)　are　respectively　obtained.　This　improves　the　result　of　X.　M.　Wang　[Commun.　Pure　Appli.　Math.,　LVII(2004),　1265-1282].　©　2007　International　Press.