The self consistent expansion applied to the factorial function

Most of the interesting systems in statistical physics can be described as nonlinear stochastic field theories. A common feature in the theoretical study of such systems is that ordinary perturbation theory seldom works. On the other hand, there exists a useful tool for the study of systems of that generic nature. That tool, the Self Consistent Expansion (SCE) is technically similar to the ordinary perturbation expansion, in the sense that it is an expansion around a solvable problem. The key point which distinguishes the SCE from an ordinary perturbation expansion, is that the small parameter of the expansion is adjustable and determined inherently by optimization of the expansion. Therefore, it allows the adaptive SCE to remain accurate relative to the inflexible ordinary expansion.