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Thread: Rearrangements of an infinite Series

  1. #1
    Mar 2012

    Rearrangements of an infinite Series

    If ak>=0 for all k in N, and Σak= prove that Σa'k= for any rearrangement Σa'k of Σak

    I think I understand this proof, I'm just having trouble making it rigorous. I was going to go the route that since ak is positive, then its sum is monotone increasing. In order to be convergent it would need to be bounded, so I can assume it's unbounded since it diverges. Then any rearrangement of the series is also going to be unbounded and then it will diverge. Is that correct thinking?
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  2. #2
    MHF Contributor
    Dec 2012
    Athens, OH, USA

    Re: Rearrangements of an infinite Series

    For a formal proof, I think you have to explicitly use the definition of series rearrangement. Here's my version:

    Rearrangements of an infinite Series-mhfseriesrearrangement.png
    Thanks from renolovexoxo
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  3. #3
    MHF Contributor

    Mar 2011

    Re: Rearrangements of an infinite Series

    to help decipher what johng has done:

    we know we can find a partial sum for the original series that is greater than any given M. this is a finite sum of terms of the original series. now in the "re-arranged" series, these terms may be scattered apart, but we can surely find a partial sum of the re-arranged series that contains ALL of them (even if we have to go out much further than we did in the original series), first we find "how far out (the largest index in the re-arragement)" we went, and include all those terms in our partial sum for the re-arranged series. this includes every term in our original partial sum, plus perhaps a lot more, but certainly no less. so this partial sum is at least the same as the partial sum of the original series (but probably a lot bigger). in any case, its bigger than M, and M can be as large as we like.

    EDIT: a note about the formalism involved. with infinite things, we can't really explicitly write them out. so, for example, to say a quantity (our series, in this instance) is infinite, we have to say something like:

    for every (finite) real number M, there exists a partial sum Sn, with Sn > M. this lets us say informally:

    $\displaystyle \sum_{k = 0}^{\infty} a_k = \infty$

    but it must be kept in mind that the "=" sign here is just a SHORTHAND for the formal definition above (which doesn't even mention infinity, because infinity is NOT a number).

    re-arranging "infinite" things can have some counter-intuitive properties (it is possible with alternating series to get finite AND infinite "re-arrangements", which justifies the cautious approach taken here), so we want to take "behavior at infinity" back to the realm of the finite whenever possible, or else the reasoning can get a little "soft".
    Last edited by Deveno; Feb 10th 2013 at 12:51 PM.
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