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Wallis sequence, Stirling’s approximation formula and some applications

  1. THE WALLIS PRODUCT FORMULA

In 1655, John Wallis wrote down the following celebrated formula:

\displaystyle\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\ldots=\lim_{n\to\infty}\prod_{k=1}^{n}\frac{4k^2}{4k^2-1}=\lim_{n\to\infty}\prod_{k=1}^n\left(\frac{2k}{2k-1}\cdot\frac{2k}{2k+1}\right)=\frac{\pi}{2}.

In other words, we have the following nice representation for \pi:

\displaystyle\pi=\lim_{n\to\infty}\frac{1}{n}\left(\frac{2\cdot 4\ldots (2n)}{1\cdot 3\ldots (2n-1)}\right)^2=\lim_{n\to\infty}\frac{1}{n}\left(\frac{(2n)!!}{(2n-1)!!}\right)^2.

This amazing product formula, derived by Wallis by a method of successive interpolation, sparked attention for centuries for many mathematicians and it provides a key ingredient in the proof of the celebrated Stirling’s approximation formula. This formula will be discussed later in this post. Moreover, in 2015 researchers from University of Rochester found an unexpected connection between Wallis’ formula and quantum mechanics. In fact, it is shown that Wallis’ formula can be derived from a variational computation of the spectrum of the hydrogen atom. For more details about this discovery, one can see the paper here. Earlier, in 2007, Wastlund published a completely elementary proof here. One quick way to find it is by using Euler’s sine representation as a infinite product:

\displaystyle\boxed{\sin x=x\prod_{n=1}^{\infty}\left(1-\frac{x^2}{\pi^2n^2}\right)}.

Taking x=\frac{\pi}{2}, one has

\displaystyle 1=\frac{\pi}{2}\prod_{n=1}^{\infty}\left(1-\frac{1}{(2n)^2}\right)=\frac{\pi}{2}\prod_{n=1}^{\infty}\left(\frac{(2n)^2-1}{(2n)^2}\right),

and thus,

\displaystyle\frac{\pi}{2}=\prod_{n=1}^{\infty}\left(\frac{(2n)^2}{(2n-1)(2n+1)}\right)=\prod_{n=1}^{\infty}\frac{4n^2}{4n^2-1}.

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