Math 2.0

statement theorem

Wallis product

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.

Explanations

L1 human

The Wallis product writes π/2\pi/2 as an infinite product of simple fractions. The partial products converge slowly to π/2\pi/2.

What this node needs for W2
  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 37 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

What depends on this

1 node uses this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Wallis_product, Proof using integration

What links here

Used by