I've got a double sum of the form
$$\sum_{k=0}^\infty \left( \sum_{i=0}^k a_{i,k} \right) $$
and I'm trying to work out how you interchange these two sums. I remember seeing a formula for this in one of my courses, but I can't remember it (nor can I find my notes). As far as I remember, it comes out as two infinite sums.
I know when the inner sum isn't in terms of $k$, you can apply Fubini-Tonelli if the inner summands are all non-negative, but here that obviously doesn't make much sense. I tried to draw a grid with the entries and count them in a different order, but I keep getting the sum indexed by $i$ on the inside so I'm a little lost.
So, if anyone could prod me in the right direction, that would be great.
Best Answer
A good way to remember is to write: $$ \sum_{k=0}^\infty \sum_{i=0}^\infty a_{i,k} \chi(i \le k), $$ where $\chi$ represents the indicator function. Then you simply interchange the summation: $$ \sum_{k=0}^\infty \left( \sum_{i=0}^k a_{i,k} \right) = \sum_{i=0}^\infty \sum_{k=0}^\infty a_{i,k} \chi(i \le k) = \sum_{i=0}^\infty \sum_{k=i}^\infty a_{i,k}.$$ Obviously, you should be concerned with when you can apply Fubini.