Reimann zeta function

I started solving a problem using the transformation of Reimann zeta function into this form: And I searched for the methods of doing this transformation and ended up with this one : It is a little bit unclear to me the way it exchanges the summation signs , the first summation changes variables from $$n \geq 1$$ to $$k \geq 1$$ : Any explanation to this , or any help on how to transform Reimann Zeta function into that form(using other methods) would be appreciated, thank you.

• There is nothing strange in this, it's just an interchange of the two sums. If you notice, the first one says "$n$ is greater or equal to $1$ and $k$ must be equal or greater to $n$". Then we may change it into "$k$ is greater or equal to $1$ provided that $n$ is equal or less than $k$". Simple reasoning. Dec 11 '18 at 9:55
• @VonNeumann so are this to expressions $\sum_{n=1}^{\infty} \sum_{k=n}^{\infty}$ and $\sum_{k=1}^{\infty} \sum_{n=k}^{\infty}$ equal , I mean am I allowed to do these kinda exchanges ? Dec 11 '18 at 9:59
• You need absolute convergence of the double-series in order to do this. (This is fulfilled here.) Dec 11 '18 at 10:24

$$\sum_{\color{red}{n}\ge 1}\sum_{\color{green}{k}\ge n}=\sum_{1\le\color{red}{ n}<\infty}\sum_{n\le \color{green}{k}<\infty}=\sum_{1\le \color{red}{n}\le \color{green}{k}<\infty}$$
where I use colors to denote the variables clearly in each sum. The notation at the LHS seems a bit sloppy, it is writing $$k\ge n$$ where $$k$$ is the variable, not $$n$$. Now choosing $$k$$ as a variable from $$1$$ to infinity we get
$$\sum_{1\le \color{red}{n}\le\color{green}{k}<\infty}=\sum_{1\le\color{green}{ k}\le\infty}\sum_{1\le \color{red}{n}\le k}=\sum_{k=1}^\infty\sum_{n=1}^k$$
where the last expression is equivalent to the sloppy form $$\sum_{k\ge 1}\sum_{n\le k}$$. This reordering is justified when the double sum is summable, this happen, by example, when it is a double sum of non-negative terms or when it is absolutely convergent.