Recently, while working on some combinatorics, I stumbled upon an interesting identity. I haven't seen it mentioned before, so I decided to share it here (with the hope that someone will find it fascinating).
The identity
holds for $$$n, m \geq 1$$$.
I encourage everyone to try to prove it by themselves before reading my proof.
Proof
Corollaries
If you have other proofs of this identity, I would gladly read about them in the comments.
are we actually gonne get this in your next div 3 ...
didn't you just switch out $$$n$$$ with $$$m$$$?
yes, but it isn't obvious (at least for me) that switching them out do not change the value of that sum.
I am probably not getting something here, but isn't that kinda the same thing as using $$$j$$$ in a for loop instead of $$$i$$$?
$$$n$$$ may differ from $$$m$$$ and the summation is over $$$k$$$.
Oh ok, I get it now. That is weird how they are equal.
why do we think the same way lol