Method of self-similar approximations

A new method is suggested to approximately find out the limit for a sequence of functions when solely several first terms of a sequence are known. The method seems to be very useful for those complicated problems of mathematical physics, statistical mechanics, and field theory in which, because of technical difficulties, one is able to calculate, using some iteration procedure or perturbation scheme, only several first approximations for quantities of interest. As an illustration, the quartic oscillator is considered, and it is shown that invoking only the terms of a perturbation theory up to the second order, the ground-state energy is defined with an accuracy of 0.1% for all values of the anharmonicity parameter, from zero up to infinity.