Zeno's paradox — Achilles and the tortoise

What just happened?

Achilles chases the tortoise. Before he reaches the finish, he must first cover half the distance, then half of what remains, then half of the rest… There are infinitely many segments, but each is half as long as the previous one. You saw that their sum does not grow to infinity — it settles exactly at the value 1.

Zeno’s paradox:

Zeno of Elea (c. 450 BC) formulated this paradox to prove — as he believed — the impossibility of motion. He argued that since Achilles must take infinitely many steps to catch the tortoise (each time he reaches the spot where the tortoise was, it has already crept a little farther), he can “never” finish. The paradox was only resolved by the mathematics of series: the sum of infinitely many, ever smaller terms can be finite.

Sum of the segments:

This is a geometric series with first term $a_1 = \tfrac12$ and common ratio $q = \tfrac12$. The partial sum after $k$ steps is $S_k = 1 - \left(\tfrac12\right)^k$ — the larger $k$, the closer to 1, but always below it.

Invariant — sum of a convergent series:

For $|q| < 1$ the geometric series is convergent, and its sum is:

Why this matters for your exams:

The sum of a convergent geometric series is a fundamental exam topic. The trick: check whether $|q| < 1$ (then the series converges), and substitute into the formula $S = a_1/(1-q)$. The same formula turns a repeating decimal into an ordinary fraction, e.g. $0.(9) = 1$.

Remember: infinitely many ever smaller terms can give a finite sum. For a convergent geometric series ($|q|<1$) the sum is $S = a_1/(1-q)$ — that is why Achilles catches the tortoise.