2,500 years ago, the Greek philosopher Zeno made this argument: "Suppose swift Achilles chases a slow tortoise, starting a little behind it. By the time Achilles reaches the tortoise's starting point, the tortoise has already moved a bit further ahead. By the time Achilles reaches that new point, the tortoise has moved further ahead again. Since this repeats endlessly, Achilles can never catch the tortoise." It sounds logical, but in reality, Achilles overtakes the tortoise with ease. So where does the argument go wrong?
Zeno's mistake was assuming that "adding infinitely many terms always makes the result infinitely large." But if the values you're adding shrink fast enough, you can add infinitely many terms and the sum will still stop at a finite value. Let's say the whole distance is 1. Achilles first covers half of it (1/2), then half of what's left (1/4), then half of what's left after that (1/8)… how far does he go if this continues forever?
The answer is exactly 1. Adding terms like 1/2 + 1/4 + 1/8 + ⋯, where each term is the previous one multiplied by a fixed ratio (here, 1/2), is called a geometric series, and as long as that ratio is less than 1, no matter how many terms you add, the sum never crosses past a certain value. Adding infinitely many terms and still settling toward a specific value like this is called "converging." Achilles catching the tortoise works exactly the same way: you can split the distance (and the time it takes) into infinitely many tiny pieces, but the sum of all those pieces is itself finite.
There's a trap here that's easy to fall into, though. Adding 1 + 1/2 + 1/3 + 1/4 + ⋯, where the denominator increases by one each time — the harmonic series — looks similar to a geometric series at first glance, but meets exactly the opposite fate. Even though each term shrinks just like before, this sum grows without bound — just extremely, extremely slowly. Even after adding a million terms, the sum is only around 14. So while it looks similar to a geometric series on the surface, "how fast the terms shrink" is what completely separates convergence from divergence.
It took over 2,000 years for Zeno's paradox to actually get a mathematical answer. It wasn't settled until Newton and Leibniz invented calculus in the 17th century, and Cauchy and Weierstrass rigorously defined the concept of a "limit" in the 19th century. The key insight is that "going through infinitely many steps" and "taking infinitely long" are two completely different things. Since time gets split up in the same proportions as the shrinking distance, you can complete infinitely many steps within a finite amount of time.
On our activity page, you can use a slider to increase the number of steps and see how close Achilles gets to the finish line, choose a ratio to see what value a geometric series converges to, and watch for yourself how the harmonic series keeps growing without bound no matter how many terms you add (just very, very slowly).