Zenos Turtle

The Greek philosopher Zeno of Elea is famous for formulating paradoxes that challenge our intuition about motion and infinity. One famous thought experiment imagines Achilles chasing a turtle. Even though Achilles runs faster, Zeno argues he can never catch the turtle: by the time Achilles reaches where the turtle was, the turtle has moved a little further ahead, and so on, ad infinitum. Formally, we can write Achilles’ path as a geometric series.

Suppose Achilles runs ten times faster than the turtle, and the turtle has a head start of distance $d$. The sequence of distances Achilles runs to reach the turtle’s successive positions is:

$d, \tfrac{d}{10},\; \tfrac{d}{100},\; \tfrac{d}{1000}, \ldots$

This is an infinite sum: $S = d + \frac{d}{10} + \frac{d}{100} + \frac{d}{1000} + \cdots$ At first glance, it seems Achilles must complete infinitely many tasks, therefore he will never reach the turtle. But the infinite series has a finite limit:

$S = \sum_{k=0}^{\infty} \frac{d}{10^k} = \frac{d}{1 - \tfrac{1}{10}} = \frac{10}{9} d.$

Achilles needs to run only a finite distance, slightly larger than $d$, to overtake the turtle. This is intuitive: The larger the speed-ratio of the both, the smaller the distance. The paradox dissolves once we accept that an infinite process can converge to a finite outcome.