I somewhat frequently run into sums with alternating positive and negative terms, where the individual terms may be much larger than the sum itself. For example, inclusion-exclusion often yields such formulas for probabilities, where the summands may be much larger than 1 although the sum is obviously between 0 and 1. Because of cancellation, a good asymptotic approximation for each term (good as in relative error is $o(1)$) does not necessarily yield a good approximation for the sum.
What are some good tricks for approximating sums like this (and where can I read about them)? I'm thinking of summation by parts as one example of what I'm looking for, but it only occasionally helps. I have a dim awareness that Fourier transforms might sometimes do something but I am not sure where to learn more about that.