I don't know if that is precisely what you are looking for. It is quite a neat formula. I came across it being used once in a proof looking at random paths where the terms all canceled neatly. However, if you had just been using as opposed to there is no way you would have seen the cancellations!
I can't quite remember what the book is - it's on my desk, but regrettably my desk is not here!
EDIT: Meant to say, I'm not actually sure if this is an example or not. I just think it is pretty, and that it is a good example of approximating a function to get much more information about it.