To understand why renormalization works you might first consider simpler situations in quantum mechanics. In this case here are explicitly solvable toy models where one can see exactly what happens and why. See my paper Renormalization without infinities - a tutorial, which discusses this in detail.
About how (or whether) one can tell whe the terms in an asymptotic series will be small see the discussions in Chapter B5: Divergences and renormalization of my theoretical physics FAQ.