How to show this "identity" for multiplication of series?

$$\sum_{k=0}^{\infty}\sum_{n=-k}^{\infty}b_{k,n} = \sum_{n=-\infty}^{\infty}\sum_{k=\max{0,-n}}^{\infty}b_{k,n}$$

Question

How to show this "identity" for multiplication of series?

$$\sum_{k=0}^{\infty}\sum_{n=-k}^{\infty}b_{k,n} = \sum_{n=-\infty}^{\infty}\sum_{k=\max{0,-n}}^{\infty}b_{k,n}$$

kainui · Answer

I'd probably play around with like finitely many of the terms and see what that looks like and then try to generalize it to an infinite case. At least I would convince myself that it was probably true even if I couldn't rigorously prove it.

anonymous · Answer

I was assuming there'd be a way to substitute something, but I would think that'd change the $b_{k,n}$ part.. But I'm not really sure how you would get a sum to go from 0 to $\infty$ to $- \infty $ to $\infty$. I figure that's what the -k part is for, but yeah.

anonymous · Answer

Im not really even sure how I'd write this out, to be honest. Should I be picking finite numbers for the indeces and then writing that out? I guess having that $-\infty$ part makes me unsure how to write it.

kainui · Answer

Yeah beats me I was sorta distracted earlier but I'll try to see.

My first attempt at visualization is to sorta sketch out a graph of the discrete points that we're summing over with coordinates (k,n) just to sorta have something to sorta picture in my mind.

kainui · Answer

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