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Concept
Wallis product
Summary
In mathematics, the Wallis product for pi, published in 1656 by John Wallis, states that
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,
See the main page on Gaussian integral.
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 :
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 .
Official source
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