h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

There's a tradition among some high school calculus teachers resolve the paradox by pointing out that an infinite series can have a finite sum. It misses an interesting and important philosophical point implied by Zeno's arguments.

On the assumption that matter, space, and time are continuous and infinitely divisible (scale invariant), we can conceive of a point-like massless particle (say, a photon) traveling at constant speed through a sequence of mirrors whose sizes and separations decrease geometrically (e.g., by a factor of two) on each step. The envelope around these mirrors is clearly a wedge shape that converges to a point, and the total length of the zigzag path is obviously finite (because the geometric series 1 + 1/2 + 1/4 + ... converges), so the particle must reach "the end" in finite time. The essence of Zeno's position against continuity and infinite divisibility is that there is no logical way for the photon to emerge from the sequence of mirrors.

The direction in which the photon would be traveling when it emerged would depend on the last mirror it hit, but there is no "last" mirror.

Achilles would never catch the tortoise because the same applies. Every time you take a snapshot of their positions the tortoise is ahead. There would be an infinite amount of these snapshots and in every one of the pictures the tortoise is ahead.

In a finish it is traditional to take a "Photo finish" at the end of the race but the race never finishes. The 'Final' photo is never taken because there is no final photo.