Wallis Product for Pi

The Wallis product is an infinite product formula for π discovered by John Wallis in 1655. It states that π/2 equals an infinite product of ratios.

Formula: π/2 = (2/1) × (2/3) × (4/3) × (4/5) × (6/5) × (6/7) × (8/7) × (8/9) × ...
General Form: π/2 = ∏(n=1 to ∞) [(2n)² / ((2n-1)(2n+1))]
Current π estimate: N/A
Terms used: 0

Product Terms

Current Values

Wallis Product: 1.000000
π/2 Estimate: 1.000000
π Estimate: 2.000000
Error from π: 1.141593

Convergence Visualization

Interactive Product Builder

π/2 = 1

How it works:

  • Each term has the form (2n)² / ((2n-1)(2n+1))
  • For n=1: (2×1)² / ((2×1-1)(2×1+1)) = 4/(1×3) = 4/3
  • For n=2: (2×2)² / ((2×2-1)(2×2+1)) = 16/(3×5) = 16/15
  • The product converges slowly to π/2

Historical Note:

John Wallis discovered this formula in 1655 while studying the area under curves. It was one of the first infinite product representations of π and provided a new way to compute π using only arithmetic operations.