I was just wondering if there was any theory or theorems involving the convergence of a function's derivative/antiderivative taken an infinite number of times. It would seem to me that there are four types of ways that it can happen: it blows up to infinity, approaches zero, approaches some function, or alternates between several functions. Based on my premliminary investigations, there's even functions which do a combination of the three. Anyway, I'm basically looking for some reading material on the subject, and was hoping that one of you people have read something about it. Furthermore, I'm looking for some reading material about techniques of evalutating/approximating integrals without a closed form, without using a sum of values in the interval as an approximation. Any help is vastly appreciated.