  In many physical problems it is not possible to find an exact solution.
However, when some parameter in the problem is small, one can obtain an
approximate solution by expanding in this parameter. This is the basis of
perturbative methods, which have been applied and developed practically in all
areas of Physics. Unfortunately many interesting problems in Physics are of
non-perturbative nature and it is not possible to gain insight on these
problems only on the basis of perturbation theory: as a matter of fact it often
happens that the perturbative series are not even convergent.
  In this paper we will describe a method which allows to obtain arbitrarily
precise analytical approximations for the period of a classical oscillator. The
same method is then also applied to obtain an analytical approximation to the
spectrum of a quantum anharmonic potential by using it with the WKB method. In
all these cases we observe exponential rates of convergence to the exact
solutions. An application of the method to obtain a fastly convergent series
for the Riemann zeta function is also discussed.
