  The main issue of this work consists in extracting one or several finite
values for the sum of series involved in perturbation theories. It is supposed
to work for all cases in which two physical parameters are involved, and makes
thorough use of dimensional arguments concerning these physical quantities.
Weak and strong coupling expansions are considered as the two faces of a formal
expression which is the central object of this method. This so-called extension
procedure is systematic. We apply it here to the divergent perturbative
expansion of the ground state energy of the anharmonic oscillator of quantum
mechanics in zero and one dimension, and provide, given a p-order divergent
expansion, an analytical expression of its estimated sum. This method which is
inspired by variational procedures provides high degree of accuracy from lower
orders of perturbation and seems to have remarkable converge properties in a
wide class of expansions concerning physical observables.
