Infinite Sums, Finite Answers

The key insight: infinitely many numbers can add to a finite total. Add 1/2 + 1/4 + 1/8 + 1/16 and keep going forever — the sum approaches exactly 1 but never overshoots. This is a convergent series: infinite steps producing a bounded, exact result.

Achilles' problem works the same way. Each catch-up interval gets shorter and shorter. The total time to overtake the tortoise is finite. Zeno's error was assuming that infinitely many steps must take infinitely long. Mathematics proved that assumption wrong.