Convergence of a double summation using diagonals

[Shawn Garsed]

Homework Statement

Show that $\sum_{k=2}^\infty d_k$ converges to $\lim_{n\to\infty} s_{nn}$.

Homework Equations

I've included some relevant information below:

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.

 

[StoneTemplePython]

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?

 

[fresh_42]

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

The question is to prove, that the summation of absolute convergent series doesn't depend on the order.

 

[StoneTemplePython]

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.