We discussed perturbations to a hamiltonian, so that H = Ho + \lambdaH'.. which makes the wavefunction and energy a function of lambda.. we then expand psi and E in a taylor series in terms of lambda.. to get the second and higher order corrections.... My question is how can we be sure that our perturbation is significantly small to allow this type of treatment. In the text it says "we assume our series converges" so this means each successive term gets smaller and smaller... In what cases can we be sure that this is a valid way to estimate the energy and when can we know we need to find a better method? It seems like if the perturbation is small we can get very close to the 'true' energy by brute force taking more and more terms, but this is not the case for large perturbations... and how do we decide what is acceptably perturbation small if we dont know the true answer???
Usually, we know roughly the magnitude of perturbation relative to the unperturbed energy. Well, if the perturbation is large, then we have to resort to some other theory.