Wallis product


In mathematics, the Wallis product for Pi|, 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 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
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 :

Relation to 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 one can deduce 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: