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I know about Riemann’s Rearrangement Theorem. But certainly you can rearrange some of the terms of a conditional series and still arrive at the same series value. For example, if you only rearrange a finite number of initial terms. This post seems to indicate that as long as terms don't move too far, you still get the same value.

  • What about rearranging terms in a double series $\sum_{n=1}^\infty\sum_{k=n}^\infty a_{n,k}$ where (for fixed $n$) $\sum_{k=n}^\infty a_{n,k}$ converges absolutely to $A(n)$, but $\sum_{n=1}^\infty A(n)$ is only conditionally convergent? Can this be rearranged to $$\sum_{n=1}^\infty\sum_{k=n}^\infty a_{n,k} \stackrel{?}{=} \sum_{k=1}^\infty\sum_{n=1}^k a_{n,k}.$$
  • Is there a collection of general theorems about the types of rearrangements that preserve the value of a conditionally convergent series?

(All I find through googling is lots of explaining Riemann's Theorem. I get that it doesn't work all the time; when does it work?!?!)

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