I was looking at the difference between an integral of a polynomial and the summation of that polynomial over the integers. Like starting with a polynomial say, x^3, over the interval 3..5 just at the integers is 3^3 + 4^3 + 5^3 = 216, but the integral over that same interval is only 136. I found the following relationship:
It's strange for a couple of reasons, why the fractions on the outside of the integral go in the pattern that they do and why does it seem to skip the 3rd derivative in every case? That makes it kind of hard to tell what the ... actually means.