Math 2.0

Number theory Theorem

Wallis product

Builds on Reduction formulaSqueeze theorem

Statement

The partial products of the following infinite product converge to π/2\pi/2: π2=∏n=1∞4n24n2−1=21⋅23⋅43⋅45⋅65⋅67⋯ .\frac{\pi}{2} = \prod_{n=1}^{\infty} \frac{4 n^2}{4 n^2 - 1} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots.

Remarks

Let In=∫0π/2sin⁡nx dxI_n = \int_0^{\pi/2} \sin^n x \, dx. The reduction formula gives I2nI_{2n} and I2n+1I_{2n+1} as explicit products. The sequence InI_n is decreasing, and I2n+1/I2n→1I_{2n+1} / I_{2n} \to 1 by the squeeze theorem. This ratio is a partial product times 2/π2/\pi. Mathlib states it as Real.tendsto_prod_pi_div_two.

History

John Wallis published the product in 1656.

Reading path

To understand this page from the ground up, read these 37 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementSqueeze theorem, statement, no evidence levelSqueeze theoremstatementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementDerivative function, definition, no evidence levelDerivative functiondefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementProduct rule, statement, no evidence levelProduct rulestatementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementHigher derivative, definition, no evidence levelHigher derivativedefinitionExtreme value theorem, statement, no evidence levelExtreme value theoremstatementContinuously differentiable function, definition, no evidence levelContinuously differen…definitionRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementSecond fundamental theorem of calculus, statement, no evidence levelSecond fundamental th…statementIntegration by parts, statement, no evidence levelIntegration by partsstatementReduction formula, concept, no evidence levelReduction formulaconceptWallis product, statement, no evidence levelWallis productstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Local maximum definition
  3. Partition of an interval definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Mesh of a partition definition
  8. Riemann sum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Riemann integrable function definition
  14. Bounded sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Continuous function definition
  18. Differentiable function definition
  19. Fermat's theorem on stationary points statement
  20. Convergent sequence definition
  21. Monotone convergence theorem for sequences statement
  22. Squeeze theorem statement
  23. Every sequence has a monotone subsequence statement
  24. Differentiable functions are continuous statement
  25. Derivative function definition
  26. Bolzano–Weierstrass theorem statement
  27. Product rule statement
  28. Boundedness theorem statement
  29. Higher derivative definition
  30. Extreme value theorem statement
  31. Continuously differentiable function definition
  32. Rolle's theorem statement
  33. Mean value theorem statement
  34. Second fundamental theorem of calculus statement
  35. Integration by parts statement
  36. Reduction formula concept
  37. Wallis product statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Wallis_product, Proof using integration
About this page
Tag0XCYA4YERS
URIm2://0XCYA4YERS
Kindstatement, theorem
WritingW1 stated
MSC11Y60
StatusThis copy of the site has no build data. Run m2 status 0XCYA4YERS.
UsesReduction formula, Squeeze theorem
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