Series for π

Adding fractions to reach a circle

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

• Leibniz: $1 - \frac13 + \frac15 - \cdots = \frac{\pi}{4}$, but convergence is slow (error ≈ 1/N).
• Basel problem: $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$.
• Conditionally convergent series can be rearranged to change their sum.

Steps

1. An astonishing sum
2. How many terms?
3. Squares of reciprocals
4. Order matters
5. Leibniz, slowly
6. Only the odd squares