Wallis Product
Build finite Wallis products and use sine-integral bounds to bracket pi from both sides.
Free theory articleBuild finite Wallis products and use sine-integral bounds to bracket pi from both sides.
Concept & interpretation
The Wallis product for π provides a sequence of rational numbers constructed from an infinite product. By evaluating powers of sine integrals and establishing recurrence relations, John Wallis originally derived a formula that converges to π/2. We accumulate the finite product Wn = ∏k=1n 4k2/(4k2-1) and use sine-integral bounds to establish a strict lower estimate (2Wn) and an upper bound (2Wn(2n+1)/(2n)) for π.
Experiment
You can interactively compute products and visualize their convergence to π in the Wallis Product visualization.
- Set the Paired factor count n to
0. The empty product is defined as 1, producing a lower estimate of 2, while the upper bound remains unavailable. - Increase the Paired factor count n to
3. The lower estimate rises to 512/175 ≈ 2.9257, and the upper bound is 256/75 ≈ 3.4133. - Continue increasing to
30. Observe the lower estimate and upper bound curve gradually enclosing π (shown in green), while the bracket width steadily shrinks below 0.052.