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Homework Help: Convergence of a double summation using diagonals

  1. Dec 17, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that ##\sum_{k=2}^\infty d_k## converges to ##\lim_{n\to\infty} s_{nn}##.

    2. Relevant equations
    I've included some relevant information below:

    Ri4oy.jpg

    3. The attempt at a solution
    So far I've managed to show that ##\sum_{k=2}^\infty |d_k|## converges, but I don't know how to move on from there.

    P.S.
    I also posted this question to math.stackexchange but haven't gotten an answer there.
     
  2. jcsd
  3. Dec 17, 2017 #2

    StoneTemplePython

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    I'm not really sure I understand what the question is?

    note: I think your author is abusing language here. The array entries (at least using matrix terminology) are better described as anti-diagonals.

    Suppose you have and ##m## x ##n## array. You can sum its entries any way you want: by row, by column, by diagonal or by anti-diagonal, or some more exotic approach -- it doesn't matter as long as you include every number once in your sum, and no numbers twice.

    Why? We're dealing with scalars in ##\mathbb R## (and possibly ##\mathbb C##) here, and scalar addition commutes in ##\mathbb R##. It could be instructive to write out the indexing of the sums for these 4 different cases.

    Some care is needed when ##\infty## is involved, but you've passed the absolute convergence test based on line 2 of your picture. With absolute convergence out of the way

    So the question is either just about indexing or commutativity of addition in reals? Or something else?
     
  4. Dec 17, 2017 #3

    fresh_42

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    The question is to prove, that the summation of absolute convergent series doesn't depend on the order.
     
  5. Dec 17, 2017 #4

    StoneTemplePython

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    yikes. I didn't read this carefully enough.

    To the extent everything is real non-negative (and not identically zero which is easy to handle) the convergent sum in line 2 is #S \gt 0##, and we can renormalize / assume without loss of generality that it sums to one. Since the anti-diagonal summing is just a re-partitioning argument (union of disjoint sets) it would seem to motivate countable additivity axiom in probability. But I don't think this is what the problem is looking for, nor is it restricted to non-negative reals.
     
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