Wallis product
In mathematics, the Wallis product for , published in 1656 by John Wallis, states that

Proof using integration 
Wallis derived this infinite product using interpolation, though his method is not regarded as rigorous. A modern derivation can be found 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

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

Now, we make two variable substitutions for convenience to obtain:

We obtain values for  and  for later use. Now, we calculate for even values  by repeatedly applying the recurrence relation result from the integration by parts. Eventually, we end get down to , which we have calculated. Repeating the process for odd values ,

We make the following observation, based on the fact that 

Dividing by :

, where the equality comes from our recurrence relation. By the squeeze theorem,

Proof using Laplace's method 
See the main page on Gaussian integral. 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 :

Relation to Stirling's approximation

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

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

See also

 John Wallis, English mathematician who is given partial credit for the development of infinitesimal calculus and pi. Viète's formula, a different infinite product formula for . Leibniz formula for , an infinite sum that can be converted into an infinite Euler product for . Wallis sieve
 The Pippenger product formula obtains e by taking roots of terms in the Wallis product. Notes

External links
 
 

Articles containing proofs
Pi algorithms
Infinite products