Wallis product

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Comparison of the convergence of the Wallis product (purple asterisks) and several historical infinite series for π. Sn is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times. (click for detail)

In mathematics, the Wallis product for π, published in 1656 by John Wallis,[1] states that

Proof using integration[edit]

Wallis derived this infinite product as it is done in calculus books today, by examining for even and odd values of , and noting that for large , increasing by 1 results in a change that becomes ever smaller as increases. Let[2]

(This is a form of Wallis' integrals.) Integrate by parts:

This result will be used below:

Repeating the process,

Repeating the process,

, from above results.

By the squeeze theorem,

Proof using Euler's infinite product for the sine function[edit]

While the proof above is typically featured in modern calculus textbooks, the Wallis product is, in retrospect, an easy corollary of the later Euler infinite product for the sine function.

Let :

[1]

Relation to Stirling's approximation[edit]

Stirling's approximation for the factorial function asserts that

Consider now the finite approximations to the Wallis product, obtained by taking the first terms in the product

where can be written as

Substituting Stirling's approximation in this expression (both for and ) one can deduce (after a short calculation) that converges to as .

Derivative of the Riemann zeta function at zero[edit]

The Riemann zeta function and the Dirichlet eta function can be defined:[1]

Applying an Euler transform to the latter series, the following is obtained:

See also[edit]

Notes[edit]

  1. ^ a b c "Wallis Formula".
  2. ^ "Integrating Powers and Product of Sines and Cosines: Challenging Problems".

External links[edit]