by Oliver Bräunling. The Leibniz formula for Pi gets more digits right when summing the first 5000 terms than when summing 6000. This happens again, e.g., summing 50,000 vs 60,000 terms.

Such unreasonably many correct digits, which then fade away again when summing more terms (only to resurface much later again), can be observed experimentally, but there is also a theoretical explanation.

The video is a homage to the paper 'Borwein, Borwein, Dilcher - Pi, Euler Numbers and Asymptotic Expansions, The American Mathematical Monthly, 1989, Vol 96, 681-687'. (for the specialists: By sweeping the error terms for the asymptotic expansion under the rug, I hope to give a more accessible and less technical explanation why these correct digits occur.)