Criteria for swapping integration and summation order
Fubini's (or Tonelli's) theorem is exactly what you need, I think, and I'm a bit disturbed to see you lump it in with a bunch of other mocking names for important mathematical ideas. Normally it's stated as a theorem about interchanging integrals with respect to measures, but for current purposes it's enough to know that both Riemann integration and summation can be expressed as integration with respect to measures.
So these theorems say that a sum and an integral $\sum_n \int f(n,x) dx$ (or more generally, several of each) can be interchanged (or rearranged) in either of the following cases:
- $f \ge 0$, or
- $\sum_n \int |f(n,x)| dx < \infty$ (and by case 1, the condition $\int \sum_n |f(n,x)|dx < \infty$ is also equivalent).
Smoothness is not enough for this to hold, essentially because smooth functions can grow rapidly at infinity. (I recommend finding yourself some counterexamples.) So you will have to find out what it is about your functions that allows this to work.