So I've been fiddling around with the Euler-Maclaurin summation formula a bit. Here's the formula: $$\sum_{i=m}^{n} f(i) = \int_{m}^{n} f(x) dx + \frac{f(n)+f(m)}{2} + \sum_{k=1}^{\lfloor p/2 \rfloor} \frac{B_{2k}}{(2k)!}(f^{(2k-1)}(n)-f^{(2k-1)}(m)) + R_{p} .$$
Usually, it yields an approximation, because the third term (the sum involving the Bernoulli numbers) and the remainder term cannot be evaluated exactly.
I wonder whether there are cases in which these terms can be evaluated, and if so, if some systematic study has been done on the circumstances under which such an evaluation is possible.