Zeno's Paradox

Zeno's Paradox refers to the following puzzle. To move any distance, one must first reach half the distance. After that, to close the remaining distance, one must first reach half of it (3/4 of the total distance). After that, to close the remaining distance, one must first yet again reach half of it (7/8 of the total distance). Et cetera. We conclude that to travel any distance at all, one must travel through infinitely many places. But a finite being cannot travel through an infinite amount of places in a finite time, so this is a paradox.

The modern non-solution

It is common for people who know some modern math to think that they know how to solve Zeno's paradox. This is because they know identities like the following:

$$1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots =\sum _{n=0}^{\infty }2^{-n}={\frac {1}{1-1/2}}=2.$$