The purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the Burgers equation. The Burgers equation is a special nonlinear problem that turns into a linear one after the Cole-Hopf transformation. Our asymptotic analysis depends on this transformation. In this paper an asymptotic approximate solution is constructed, which is given by the inverse Cole-Hopf transformation of a summation of n heat kernels. The k-th order moments of the exact and the approximate solution are contracting with order O((root t)(k-2n-1+1/p)) in L(p)-norm as t -> infinity. This asymptotics indicates that the convergence order is increased by a similarity scale whenever the order of controlled moments is increased by one. The theoretical asymptotic convergence orders are tested numerically. (C) 2010 Elsevier Inc. All rights reserved.