Zeno’s Paradox

Infinite sums with finite answers

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• An infinite sum can have a finite value, like $\tfrac12 + \tfrac14 + \cdots = 1$.
• $\sum ar^k = \frac{a}{1-r}$ when $|r| < 1$. Otherwise the sum diverges.
• $0.999\ldots = 1$ exactly.

Steps

1. Can you reach the wall?
2. Filling a square
3. Infinite geometric series
4. Sum to infinity
5. Thirds
6. A quarter, and a quarter of a quarter
7. Point nine recurring