Interchanging order of summation mechanically How can I interchange order of summation mechanically, without thinking? For instance, I had to interchange the sums below (assume $i$ is a constant where $i\gt 0$).
$$\sum_{n\ge 1}\sum_{i\lt k \lt n} a_{nk}$$
I wrote down a matrix, and read it by columns instead of rows, arrived at:
$$\sum_{k\gt i}\sum_{n\gt k} a_{nk}$$
Which I think it's correct. 
But there must be a way to do it mechanically, without writing down the matrix. By mechanically, I mean something like what happens with this sum, where I can make the steps without thinking:
$$\sum_{n\gt k}\frac{1}{n} z^n = \sum_{n-k\gt k-k}\frac{1}{n} z^n $$
$$m=n-k\implies n=m+k$$
$$\sum_{n-k\gt 0}\frac{1}{n} z^n = \sum_{m\gt 0}\frac{1}{m+k} z^{m+k}$$
No thinking required here, just algebra.
Concrete Mathematics calls my example a "rocky road" double summation and suggests to use Iverson notation, but I haven't been able to apply it to my case.
 A: Provided that the constant $ i$ is at least $-1$, we have
$$ n\ge 1\;\land \;i<k<n\iff k>i\;\land\;n>k$$
Therefore (provided rearranging is justified because either we have absolute convergence or because instead of a series running to $\infty$ we really have a sun runnin to some $N$, say)
$$ \sum_{n=1}^\infty\sum_{k=i+1}^{n-1}=\sum_{k=i+1}^\infty\sum_{n=k+1}^\infty$$
A: I use this way:
$$\begin{align*}\\\sum_{n=1}^{\infty} \sum_{k=i + 1}^{n - 1} a_{n, k} &= \sum_{n=1}^{\infty} \sum_{k=1}^{\infty} a_{n, k} \cdot 1(i < k < n)\\&= \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} a_{n, k} \cdot 1(i < k < n)\\&= \sum_{k=i + 1}^{\infty} \sum_{n=1}^{\infty} a_{n, k} \cdot 1(k < n)\\&= \sum_{k=i + 1}^{\infty} \sum_{n=k+1}^{\infty} a_{n, k}\end{align*}$$
A: I preassume that $i$ is some fixed nonnegative integer. 
Let $b_{nk}=1$ if $n\geq1\wedge i<k<n$ and $b_{nk}=0$ otherwise. 
Then: $$\sum_{n\geq1}\sum_{i<k<n}a_{nk}=\sum_{n\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}a_{nk}b_{nk}=\sum_{k\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}a_{nk}b_{nk}=\sum_{k>i}\sum_{n>k}a_{nk}$$
This of course if the sum is well-defined. No matrix needed.
