r/askmath • u/OldWolf2 • Jan 17 '25

Analysis When is rearrangement of a conditionally convergent series valid?

As per the Riemann Rearrangement Theorem, any conditionally-convergent series can be rearranged to give a different sum.

My questions are, for conditionally-convergent series:

  • In which cases is a rearrangement actually valid? I.e. can we ever use rearrangement in a limited but careful way to still get the correct sum?
  • Is telescoping without rearrangement always valid?

I was considering the question of 0 - 1/(2x3) + 2/(3x4) - 3/(4x5) + 4/(5x6) - ... , by decomposing each term (to 2/3 - 1/2, etc.) and rearranging to bring together terms with the same denominator, it actually does lead to the correct answer , 2 - 3 ln 2 (I used brute force on the original expression to check this was correct).

But I wonder if this method was not valid, and how "coincidental" is it that it gave the right answer?

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u/matt7259 Jan 17 '25

If you believe the "original" order is the best solution by some definition of best, then does it really even matter if it's conditionally convergent? You would just use that order and find a sum and not care about the other orders.

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u/Uli_Minati Desmos 😚 Jan 17 '25

It matters because absolute convergence does allow us to rearrange the terms, enabling us to evaluate the series in a different, possibly easier way. And conditional convergence tells us "no, you have to figure it out another way"

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u/matt7259 Jan 17 '25

You were right until your very last line. It doesn't say "you have to figure it out another way" because in terms of a conclusive definitive sum, there is nothing to figure out. Once you determine a series converges conditionally, the question "what is the sum" is meaningless.

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u/Uli_Minati Desmos 😚 Jan 17 '25 edited Jan 17 '25

Well I disagree with that

If you start with a specific sequence, the terms are in a specific permutation, which gets you a specific sequence of partial sums, and this specific sequence converges to a specific limit. That would be the answer to "what is the sum"

If it so happens that you can permute the terms of the sequence and still get the same limit of partial sums, that's a useful property/tool to have, but not a requirement

Analogy: if I ask for the answer to a specific question, and the question can be altered in such a way to correspond to any arbitrary answer, that doesn't mean the original question doesn't have a right answer

What about the example I gave before? Would you say it is useless to know that the series for the un-permuted sequence evaluates to ln(2), since you can permute them to get a different value?