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I started solving a problem using the transformation of Riemann zeta function into this form:

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And I searched for the methods of doing this transformation and ended up with this one:

enter image description here

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 $k \geq 1$ :

enter image description here

Any explanation to this, or any help on how to transform Riemann Zeta function into that form (using other methods) would be appreciated, thank you.

Another User
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Maths Survivor
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    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. – Enrico M. Dec 11 '18 at 09:55
  • @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 ? – Maths Survivor Dec 11 '18 at 09:59
  • You need absolute convergence of the double-series in order to do this. (This is fulfilled here.) – p4sch Dec 11 '18 at 10:24

1 Answers1

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Something that works very nice to see the change of indexes is writing a sum like this

$$\sum_{\color{red}{n}\ge 1}\sum_{\color{green}{k}\ge n}=\sum_{1\le\color{red}{ n}<\infty}\sum_{n\le \color{green}{k}<\infty}=\sum_{1\le \color{red}{n}\le \color{green}{k}<\infty}$$

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.

Masacroso
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