All the Way Down
12 days ago
- #Archimedes
- #infinite series
- #mathematics
- The infinite series 1/4 + 1/16 + 1/64 + 1/256 + … sums to 1/3.
- Archimedes discovered this summation around 200 BC.
- Visual proof using a unit square: largest black square has area 1/4, next 1/16, etc., totaling 1/3 when combined.
- Another visual proof uses triangles: largest black triangle has area 1/4, next 1/16, etc., also totaling 1/3.
- Both proofs show black, white, and gray regions occupying equal areas, confirming the sum.