Wallis partial products approaching π/2
Wallis product: (2/1)(2/3)(4/3)(4/5)(6/5)(6/7)... The partial products converge to π/2 ≈ 1.5708 from below, oscillating around the limit.
The Wallis product: alternating even fractions
π/2 = (2/1)·(2/3)·(4/3)·(4/5)·(6/5)·(6/7)·…
= Π_{n=1}^∞ (4n²)/(4n²−1)
Wallis derived this in 1655 by comparing integrals of powers of sin(x). It was the first product formula for π.
Integrals of sin^n(x) from 0 to π/2: even/odd pattern produces Wallis
Even n: I(n) = (π/2)·(1/2)·(3/4)·(5/6)…(n−1)/n. Odd n: I(n) = 1·(2/3)·(4/5)…(n−1)/n. The ratio of adjacent integrals I(2n)/I(2n+1) → 1, giving the Wallis product.
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