Interchanging discrete summation signs

[tgt]

When needed to do that, I found it much easier to pretend it's an integral summation and then draw the area diagram then work it out from the picture the new terminals for the integral. Then convert that back into the discrete sum. Is that how you would do it?

However for three or more variables that method could be tough. How would you do it then? Both for discrete and continuous cases.

 

[quasar987]

Are we talking about an infinite sum or a finite sum?

For the infinite case, Fubini's theorem for the Lebesgue integral applied to the counting measure gives conditions for when interchanging summation sign is legal.