| Achilles runs at 1 unit/s and Tortoise at x < 1 unit/s. Tortoise gets a head start of 1 unit. How long until Achilles catches up? Well first, Achilles has to travel 1 unit because Tortoise got a head start. That takes 1s. By then, Tortoise managed to run x units, so Achilles has to catch-up, which takes another x seconds, and by then... etc: The time it takes Achilles to catch up is 1 + x + x^2 + ... Since x < 1, this converges and the answer is 1/(1-x). It takes 1/(1-x) seconds for Achilles to catch up. Plot the lines: Achilles runs along A(t) = t, the tortoise T(t) = 1 + x t. Solving for t: t = 1 + x t = t (1 - x) = 1; t = 1/(1-x). Ok, now what if x > 1? Let's use x = 2. 1/(1-2) = -1. It takes -1 seconds for Achilles to catch up. Don't believe me? Plot it: A(t) = t, T(t) = 1 + 2t; Achilles catches up with Tortoise 1 second before the race starts! There you have it, a physical interpretation of why 1 + 2 + 4 ... = -1. (I have no example right now for 1 + 2 + 3 ..., sorry.) So you see, this isn't nonsense, though I admit it DOES depend on definitions! |