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Lets say I have a double infinite sum $\sum\limits_{n=0}^\infty a_n\sum\limits_{k=n}^\infty b_k$ If I know that the sum is absolutely convergent and that $\sum\limits_{k=n}^\infty b_k$ is absolutely convergent, what kind of rearrangements are valid? For example I would like expand the series and collect certain terms into meaningful groups, is that a "legal" operation provided all the series I work with are absolutely convergent?

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If the series $\sum_{n=1}^\infty |a_n| \sum_{k=n}^\infty |b_k| $ converges, then you can do whatever you want with $a_n b_k$; any rearrangement will converge to the same sum. More generally: if $I$ is an index set and $\sum_{i\in I} |c_i|<\infty$, then any rearrangement of $c_i$ converges to the same sum. This is because for every $\epsilon>0$ there is a finite set $S\subset I$ such that $\sum_{i\in I\setminus S}|c_i|<\epsilon $; any method of summation will use up $S$ at some point, and after that the sum is guaranteed to be within $\epsilon$ of $\sum_{i\in S} c_i$.

But if you only know that $$\sum |b_k|\tag{1}$$ and $$\sum_{n=1}^\infty \left|a_n \sum_{k=n}^\infty b_k \right|\tag{2} $$ converge, then rearrangements can go wrong. For example, take the series $\sum b_k $ to be $\frac{1}{2} -\frac{1}{2}+\frac{1}{4}-\frac{1}{4}+\frac{1}{8}-\frac{1}{8}+\dots$ then every other sum $\sum_{k\ge n} b_k$ is zero, and the corresponding $a_n$ could be arbitrarily large without disturbing the convergence of (2). In this situation you could even have infinitely many terms $a_nb_k$ that are greater than $1$ in absolute value; clearly this series can't be rearranged at will.

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