Wallis product

In mathematics, the Wallis product for , published in 1656 by John Wallis, states that
Proof using integration.
Wallis derived this infinite product as it is done in calculus books today, by examining formula_2 for even and odd values of formula_3, and noting that for large formula_3, increasing formula_3 by 1 results in a change that becomes ever smaller as formula_3 increases. Let
(This is a form of Wallis' integrals.) Integrate by parts:
This result will be used below:
Repeating the process,
Repeating the process,
By the squeeze theorem,
Proof using Euler's infinite product for the sine function.
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 formula_21:
Relation to Stirling's approximation.
Stirling's approximation for the factorial function formula_23 asserts that
Consider now the finite approximations to the Wallis product, obtained by taking the first formula_25 terms in the product
where formula_27 can be written as
Substituting Stirling's approximation in this expression (both for formula_29 and formula_30) one can deduce (after a short calculation) that formula_27 converges to formula_32 as formula_33.
Derivative of the Riemann zeta function at zero.
The Riemann zeta function and the Dirichlet eta function can be defined:
Applying an Euler transform to the latter series, the following is obtained: