The first part of the talk is a review of analytical estimates of the asymptotic properties of formal series, as they appear in the normal form method with Lie series and in the direct method of determination of formal integral series. The application of such methods in realistic physical problems allows to obtain useful estimates of the size of the region of Nekhoroshev stability and of the time of practical stability of the orbits. he second part of the talk presents improved asymptotic estimates of the size of the remainder and of the optimal order of truncation of formal series, based on a step by step comparison f the predictions of theory with numerical results obtained by calculating formal integral series in model 2D isochronous hamiltonian systems up to a high order.