Mathematics

The beautiful lie: 1+2+3+... = -1/12

Add every whole number, 1+2+3+4 on forever, and the running total races to infinity, so the series has no ordinary sum. Yet a famous equation writes 1+2+3+... = -1/12. Honestly read, -1/12 is not the total but the finite fingerprint left after the infinite part is stripped, and the same number predicts the measured Casimir force between two metal plates.

Sources & further reading

What's new (Terence Tao's mathematics blog)The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuationA Fields-medalist mathematician's careful explainer showing -1/12 is the leftover constant when you smoothly add up 1+2+3+..., not the actual total, which is infinite.NIST Digital Library of Mathematical FunctionsDLMF 25.6: Riemann Zeta Function, Integer ArgumentsThe official US government mathematics reference giving the exact value zeta(-1) = -1/12, defined by continuing the zeta function past where its sum works.Physical Review Letters (American Physical Society)Demonstration of the Casimir Force in the 0.6 to 6 micrometre RangeThe 1997 experiment that measured the tiny attractive force between two metal surfaces predicted by regularizing an infinite sum of vacuum energies.Physical Review Letters (American Physical Society)Precision Measurement of the Casimir Force from 0.1 to 0.9 micrometreA 1998 follow-up that measured the same vacuum-energy force to about 1%, tightly confirming the effect behind the regularized calculation.Cambridge University PressString Theory, Volume 1: An Introduction to the Bosonic StringThe standard graduate textbook where the same -1/12 determines the 26 dimensions in which the simplest string theory is mathematically consistent.American Mathematical Society / London Mathematical SocietyRamanujan: Letters and Commentary (History of Mathematics, vol. 9)A scholarly AMS/LMS volume of Ramanujan's actual letters, the documented origin of the 1+2+3+... = -1/12 claim.