Can perturbation series oscillate in magnitude and still be useful?

[kurious]

Perturbation theory provides good answers as long as the contributions get smaller and smaller as we go to higher and higher orders. Then we only need to compute the first few diagrams to get accurate results.
Is it possible for a perturbation series to get bigger then smaller then bigger-it wouldn't have a finite sum unless some terms were negative, but could such a perturbation expansion be useful in physics?

 

[marlon]

A perturbation theory as you envision it cannot come from expansions in coupling-constants. So my answer is no.

Remember that the contributions will always get smaller and smaller in higher orders, this is exactly what the expanion is about.

The criterium for deciding whether one can or cannot use expanions is the fact that the coupling constant must be relativly small, just like the conditions needed for calculation a Taylor-series...


The way you suggest a finit theory through an alternating expansion is useless because the effects per order would cancel out each other.