Switch place of 2 infinite summations In our course about queuing theory sometimes this rule is used:  
$$\sum_{n=0}^\infty \sum_{k=n+1}^\infty n\cdot k = \sum_{k=1}^\infty \sum_{n=0}^{k-1} n\cdot k$$
Another example (but without infinity):
$$\sum_{n=0}^m n \sum_{i=0}^{n-1} i = \sum_{i=0}^{m-1} i \sum_{n=i+1}^{m} n$$ for a given m.
These are just 2 examples but similar operations like these are used a few times and I can't seem to figure out what the general rule is. I also couldn't find it on the wikipedia page about summation (http://en.wikipedia.org/wiki/Summation).
Could someone explain/prove the rule or point me to a link where I can find information about it?
 A: It's more of a process than a rule, but here's how I would do it. Take your first sum $\sum_{n=0}^\infty \sum_{k=n+1}^\infty n\cdot k$, and look through first what values of $k$ does appear at all. Since the lowest value of $k$ anywhere in the sum is $1$, and $k$ is unbounded above, the outer summation has to be $\sum_{k=1}^\infty$.
Now we look at a given value of $k$, and see what values of $n$ permits that $k$ to appear in the original sum. We see that this happens when $k\geq n+1$ which means that $n\leq k-1$, so $k-1$ has to be the upper limit of the inner sum. The value $n=0$ is permitted by any value of $k$, and it is the lowest value of $n$ appearing, and therefore has to be the lower limit of the sum. The inner sum is therefore $\sum_{n=0}^{k-1}$.

I figured an illustration by picture might come in handy as well. Assume we have some function $f(n, k)$ (in your case $f(n, k) = nk$), and we want to find the sum
$$
\sum_{n=0}^\infty \sum_{k=n+1}^\infty f(n, k)
$$
Then the picture below is a picture of (a small part of) the first quadrant of the $nk$-plane (including the axes). The brown dots represents the points whose function value you are interested in:

Now, this picture is taken out of its original context, but works fine, assuming we have $n$ along the $x$-axis, and $k$ aling the $y$-axis. Then we see that if we want $k$ to be the first index of summation, we sww that it goes from $1$ to $\infty$, and that for each $k$, $n$ goes from $0$ to $k-1$. This way of drawing a picture of the terms you're interested in always works, also with finite sums. It's also a big help if you're ever going to do the same for integration.
A: As long as your series is absolutely convergent you are free to add up the terms in any order you want.  You have to make sure you get them all, and only once.  In your example, you are summing over all pairs $(n,k)$ with $k \gt n$.  You can either pick $n$ first and add up all the larger $k$, or $k$ first and add up all the smaller $n$.  The first is the left side, the second is the right.
