Changing the order of different kinds of sums

[Mandelbroth]

I'm having trouble understanding this.

Suppose I have a sum ##\displaystyle \sum_{i=1}^{n}\left[\int_{a}^{b} f(t) dt\right]##, where f(t) depends on both n and i. Under what conditions could this expression be equal to the same expression with the integral and the summation in reversed order? That is, I want to know when I can say that $$\sum_{i=1}^{n}\left[\int_{a}^{b} f(t) dt\right] = \int_{a}^{b}\sum_{i=1}^{n}\left[f(t)\right]dt$$

A math friend of mine mentioned something about Fubini's theorem, but I can't see how it applies...

All help is greatly appreciated.

 

[mathman]

For a given finite n, the sum and integral can be switched. Fubini's theorem has to do with double integrals.

In case the upper limit of the sum is infinite you do need some justification to switch.

 

[micromass]

You have no problem here because your sum is finite. If your sum is infinite (thus if it is a series), then you need justification. Fubini's theorem is one such theorem which says when you can switch sum and integral. Questions like these are answered in measure theory.

Other useful theorems are the monotone and dominated convergence theorems.