Suppose we have the following problem:
$$\sum_{d=0}^{\infty} \sum_{s=0}^d f(s)g(d)$$
where $f$ and $g$ are two arbitrary function. If I want to sum over $s$ first and then $d$ what changes do I have to make? Is it enough to do
$$\sum_{s=0}^{\infty} \sum_{d=0}^{\infty} f(s)g(d)$$
I feel as if we would be missing something if we do this. Is this correct or do we need to add in a factor to keep track of how many times each pair $(s,d)$ occurs?
The correct interchange is
$$\sum_{d=0}^\infty\sum_{s=0}^d=\sum_{s=0}^\infty\sum_{d=s}^\infty$$
The key is, $s$ is never greater than $d$.