1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:


    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.

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


    User Avatar
    Science Advisor
    Gold Member

    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


    User Avatar
    2017 Award

    Staff: Mentor

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


    User Avatar
    Science Advisor
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted