MPEJ Volume 3, No.5, 25pp Received: Mar 18, 1997, Revised: Jul 28, 1997, Accepted: Oct 1, 1997 Antonio Giorgilli, Ugo Locatelli On classical series expansions for quasi-periodic motions ABSTRACT: We reconsider the problem of convergence of classical expansions in a parameter $\epsilon$ for quasiperiodic motions on invariant tori in nearly integrable Hamiltonian systems. Using a reformulation of the algorithm proposed by Kolmogorov, we show that if the frequencies satisfy the nonresonance condition proposed by Bruno, then one can construct a normal form such that the coefficient of $\epsilon^s$ is a sum of $O(C^s)$ terms each of which is bounded by $O(C^s)$. This allows us to produce a direct proof of the classical $\epsilon$ expansions. We also discuss some relations between our expansions and the Lindstedt's ones.