Inutuitive

Perturbation theory is a large collection of iterative methods for obtaining approximate solutions to problems involving a small parameter $\epsilon$. These methods are so powerful that sometimes it is actually advisable to introduce a parameter $\epsilon$ temporarily into a difficult problem having no small parameter, and then finally to set $\epsilon =1$ to recover the original problem. This apparently artificial conversion to a perturbation problem may be the only way to make progress.

The thematic approach of perturbation theory is to decompose a tough problem into an infinite number of relatively easy ones. Hence, perturbation theory is most useful when the first few steps reveal the important features of the solution and the remaining ones give small corrections.

Advanced mathematical methods for scientists and engineers by Bender and Orszag

Concrete

Abstract

Why is it interesting?

source: http://faculty.washington.edu/acherman/PCM_Resurgence.pdf