Wallis Product
Build finite Wallis products and use sine-integral bounds to bracket pi from both sides.
By Riano · Published · Updated · Report a correction
Wallis Product
Concept Overview
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.
Let Im = ∫0π/2 sinm(t) dt. Integration by parts gives Im = ((m-1)/m) Im-2, with I0=π/2 and I1=1. Multiplying the even and odd reductions yields I2n+1/I2n = 2Wn/π. For n≥1, the strict inequalities I2n+1 < I2n < I2n-1 imply 2Wn < π < 2Wn(2n+1)/(2n). The upper formula does not apply at n=0.
Worked Example
Consider evaluating the product and bounds for n = 4 paired factors.
- Calculate W4 = ∏k=14 4k2 / (4k2 - 1). The terms are (4/3) · (16/15) · (36/35) · (64/63).
- Multiply the terms: W4 = 16384 / 11025 ≈ 1.486077.
- The lower estimate for π is 2 · W4 = 32768 / 11025 ≈ 2.972154.
- The upper bound for π relies on multiplying the lower estimate by (2(4)+1)/(2(4)) = 9/8. So, 32768 / 11025 · (9/8) = 36864 / 11025 ≈ 3.343673.
- The bracket width is the difference between upper and lower bounds: (36864 - 32768) / 11025 = 4096 / 11025 ≈ 0.371519.
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.
Limitations
The numerical convergence of the Wallis product is relatively slow; computing the finite product even up to 30 pairs leaves a bracket width around 0.05. While mathematically robust, this method is inefficient for high-precision decimal calculations of π compared to Machin-like formulas or Ramanujan's algorithms. Additionally, at n = 0 (the empty product), the upper bound formula is unavailable since it involves a division by zero in its bounding fraction (2n+1)/(2n).
References
The recurrence relation and the rigorous construction of the finite lower and upper bounds using sine-power integrals are analyzed in Sigurd Angenent's notes:Wallis' Product for π.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Wallis Product module.
Try Wallis Product on Riano →