I am trying to exchange the order of a double summation, but I think I've failed on doing it. The sum and my result is
$\sum_{r=0}^n\sum_{m=r}^\infty=\sum_{m=0}^\infty\sum_{r=0}^m$
I obtained this drawing a triangle and seeing the behavior of the indices, but I'm almost sure that there is an error, since the main result of the whole equation is not satisfactory. So I would be really grateful if some could help me to find the way to solve this. Thanks in advance.
As @rhys suggested. The way of change the order of the double sum is taking into account that we are performing a summation over an infinite triangle $0 \leq r \leq m $. So the double sum turns into
$\sum_{r=0}^n \sum_{m=r}^\infty = \sum_{m=0}^n \sum_{r=0}^m + \sum_{m=n+1}^\infty \sum_{r=0}^n $