Infinite Paradoxes

Where intuition breaks

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Divergent series can’t be summed, and regrouping their terms gives contradictions.
• Gabriel’s horn: finite volume, infinite surface area.
• Rearranging a conditionally convergent series can make it add up to any target.

Steps

1. Grandi’s series
2. Gabriel’s horn
3. Thomson’s lamp
4. Rearrange to order
5. Achilles and the tortoise
6. The first climb