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.<p>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.<p>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.)