It seems that with continuous functions, we have in calculus an apparatus for "short cutting" an infinite sum. However, when we move to the discrete case, it seems that we don't have the equivalent theoretical machinery.
What is it about sums that makes them so much more intractable than their associated integrals? In other words, why isn't there a fundamental theorem of infinite sums like there is for calculus?
A possible alternative to calculus is considering measure theory. I realize this might be outside of the scope of the question, but measure theory allows us to discuss usual integrals and sums to be one in the same.