Formula: π = 4(1 - 1/3 + 1/5 - 1/7 + ...)
Convergence: ~0.1 digits per term
Description: Derived from the Taylor series expansion of arctan(x). When x=1, arctan(1) = π/4, giving us this elegant but very slowly converging series.
Formula: π = √12(1 - 1/(3·3) + 1/(5·3²) - 1/(7·3³) + ...)
Convergence: ~0.6 digits per term
Description: Discovered by the Kerala school in India centuries before similar European work. This method converges much faster than Gregory-Leibniz.
Formula: 1/π = (2√2/9801) Σ (4n)!(1103+26390n)/(n!)⁴·396⁴ⁿ
Convergence: ~8 digits per term
Description: Derived by mathematical genius Srinivasa Ramanujan with little formal training. Each term provides about 8 additional digits of accuracy.
Formula: 1/π = 12 Σ (-1)ⁿ(6n)!(13591409+545140134n)/((3n)!(n!)³·640320³ⁿ⁺³⸍²)
Convergence: ~15 digits per term
Description: The most efficient known formula, used for modern record-setting calculations of trillions of digits of π.
| Method | Era | 1 Digit | 5 Digits | 10 Digits | 15 Digits | Efficiency (digits/term) |
|---|