And note that S(a) - S(b) should be [f(a) - f(b)] + [f(a + 1) - f(b + 1)] + [f(a + 2) - f(b + 2)] + …, which may converge even if neither S(a) nor S(b) converge in themselves.
This derivative tells us a lot! Integrating its derivative tells us everything about what S should be, except for an unknown +C term.
But how do we figure out what the +Cs in the integrations should be taken to be?
So let f(x) = x^p.
At every other power, everything works out smoothly, to some finite number.
Well, the defining property of the factorial is that x! = x * (x - 1)!. In other words, log(x!) = log(x) + log((x - 1)!).