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I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are.
As far as I know $\pi$ comes from geometry (although it does have an equivalent analytical definition), and $e$ comes from calculus.
I cannot see any reason why they should be linked and the proof doesn't really give any insights as to why the equation works.
Is there some nice way of explaining this?