Pi's Slowest Sum Fails on Purpose: 1, -1, 5

The Leibniz series for pi—1 minus a third plus a fifth minus a seventh, forever—is famous for being useless: it needs about ten times as many terms for each new correct digit. But do something nobody bothers to: add exactly 500,000 terms, then subtract the result from the true pi and look at the mistakes. They are not random. Most digits are perfectly correct; the errors sit at evenly spaced positions, and their values are 2, -2, 10, -122—which is exactly twice the Euler numbers 1, -1, 5, -61, the same sequence that appears in the power series for the secant. This isn't a coincidence: the error of the series has an exact asymptotic expansion whose coefficients ARE the Euler numbers, and because twice 500,000 is a clean power of ten, each Euler number lands in its own block of digits (with negatives showing up as a borrow, a run of nines). The Euler numbers count zigzag permutations—the same up-down-up-down rhythm as the series itself. The real lesson: when an approximation is wrong, the way it's wrong has structure, often richer than the answer. Every value here is computed to 80 digits with mpmath and verified—the expansion recovers 1, -1, 5, -61, 1385 exactly. Chapters 0:00 The Slowest Formula for Pi 0:54 How Slow Is Slow 2:30 Sum Exactly Half a Million 3:17 Line Them Up 5:01 The Wrong Digits Are a Sequence 6:05 Those Are the Euler Numbers 7:15 Why the Error Is Structured 8:59 Why They Fall Into Blocks 10:20 What the Euler Numbers Are 11:22 Look at What Is Left Over Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/a...