One of my favorite proofs that the sums of each row are powers of two comes from the fact that the numbers in row n+1 are the coefficients of the powers of (a+b)ⁿ, so setting a=b=1 you get 2ⁿ (most discrete math students seeking to prove this end up reaching for induction which is a heavier proof than this).
https://nonagon.org/ExLibris/paul-erdos-bertrands-conjecture
That's beautiful. I wonder why the -2 is there, though. To fix it, we would need
π = -x + 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 •••
where x = 2, and so it would be
π = -1/½ + 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 •••
which makes the -1st triangle number ½, I guess.
π = 3 + 2/3 (1/4 − 1/20 + 1/56 - ...)
lixtra•8mo ago
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