Forever, Slowly

The harmonic series

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

• Terms shrinking to zero is necessary for a sum to converge, but not enough.
• The harmonic series $\sum \frac1n$ diverges (Oresme's grouping argument).
• $\sum \frac{1}{n^2}$ converges, to $\frac{\pi^2}{6}$.

Steps

1. Shrinking terms
2. Oresme’s proof
3. Watch it crawl
4. Leaning tower of books
5. The harmonic staircase
6. Passing three
7. Squares converge