The Wallis product writes π/2 as an infinite product of simple fractions: (2/1) × (2/3) × (4/3) × (4/5) × (6/5) × (6/7) × ⋯ Each even number appears twice, once larger and once smaller than its neighbours. Multiply enough terms and the product converges to π/2 ≈ 1.5708.
After n pairs of factors (2/1·2/3), (4/3·4/5), … the product climbs toward π/2 ≈ 1.5708 (red line). The convergence is monotone from below. After 15 pairs the product reaches 1.537, still 2% short.
John Wallis derived this formula in 1655 from the integral ∫₀^(π/2) sinⁿ(x) dx, comparing the cases of even and odd n. What makes it remarkable is that it derives π from pure multiplication of rational numbers, with no geometry involved. The same product emerges from the Gamma function identity: π = Γ(1/2)².
The Wallis product converges very slowly: after n pairs the error is of order 1/(4n). It has enormous theoretical importance as one of the first infinite products ever studied, opening the path to the analysis of sin(x) = x∏(1 - x²/n²π²) and the entire theory of infinite products in complex analysis.
Wallis studied ∫₀^(π/2) sinⁿ(x) dx. The even and odd cases give slightly different values, and their ratio (using reduction formulas) produces the Wallis product. The curves of sinⁿ(x) narrow as n increases: their total areas form a pattern that locks in π/2.