Talk:Wallis product

message: Suppose we write
:W_n=\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdots\frac{2n}{2n-1}\cdot\frac{2n}{2n+1}
Then
:
\begin{align}
\ln(W_n)&=\ln\left(\frac{2}{1}\right) + \ln\left(\frac{2}{3}\right) + \ln\left(\frac{4}{3}\right) + \ln\left(\frac{4}{5}\right) + \ln\left(\frac{6}{5}\right)+\dots + \ln\left(\frac{2n}{2n-1}\right) + \ln\left(\frac{2n}{2n+1}\right)\\
&=\ln\left(\frac{2}{1}\right) - \ln\left(\frac{3}{2}\right) + \ln\left(\frac{4}{3}\right) - \ln\left(\frac{5}{4}\right) + \ln\left(\frac{6}{5}\right) - \dots + \ln\left(\frac{2n}{2n-1}\right) - \ln\left(\frac{2n+1}{2n}\right)\\
\end{align}
where, in the last line, all the logs are positive.
For the sum of the magnitudes of these terms, we have
:
\sum_{r=1}^{2n} \ln\left(\frac{r+1}{r}\right)>\int_{1}^{2n}\ln\left(\frac{x+1}{x}\right)\,dx=2n\ln\left(\frac{2n+1}{2n}\right)+\ln\left(\frac{2n+1}{4}\right)
Both of the terms in the result of the integration are positive and the second clearly diverges as n \rightarrow \infty ; so, therefore, must our sum of magnitudes.  \ln(W_n) is thus conditionally convergent.  By suitably ordering its terms, we can therefore make it converge to any real value and it follows that, by ordering its factors, W_n can be made to converge to any positive real number.  Any proof of the Wallis formula which fails to take into explicit account the order of the factors would thus appear to be spurious.
IanHH (talk) 16:40, 25 July 2008 (UTC)
thread_title: Order of Factors

message: :Using zeta function regularization,
::-\zeta'(s)=\sum_{n=1}^\infty \frac{\ln n}{n^s}
:Plugging in s=0 gives:
::-\zeta'(0)=\frac{1}{2} \ln\left(2 \pi\right)=\sum_{n=1}^\infty \ln n=\ln \prod_{n=1}^\infty n=\ln(\infty!)
::\Rightarrow \infty!=\sqrt {2 \pi}
This is wrong. It is wrong because the series definition of \zeta'(s) is divergent at 0. Therefore, it cannot be used to calculate the value of \zeta'(0) and is not equivalent to its actual value, \frac{1}{2} \ln\left(2 \pi\right). In other words, \frac{1}{2} \ln\left(2 \pi\right)\neq\sum_{n=1}^\infty \ln n.  — Preceding unsigned comment added by 173.80.198.94 (talk) 01:36, 3 January 2012 (UTC)
thread_title: Infinity Factorial

message: You say "Wallis' product is, in retrospect, an easy corollary of the later Euler formula for the sine function", and then "Proof using Euler's infinite product for the sine function".  Would it be more accurate to say "sinc function" in each case, because you're using the function "sin(x)/x", not just "sin(x)"?
Ross Finlayson (talk) 23:24, 20 April 2018 (UTC)
replies: text: The two are essentially equivalent (each can be trivially derived from the other), so I think it's probably fine as is.  –Deacon Vorbis (carbon • videos) 00:25, 21 April 2018 (UTC)
replies: text: Yes, it's probably 'fine' as is - but is it 'best' as is?  E.g., the title says "Proof using Euler's infinite product for the sine function", but - strictly speaking - the infinite product is for the sinc function, not the sine function.  Would it not be marginally clearer to replace "sine" with "sinc" in both of the cases that I noted? Ross Finlayson (talk) 01:17, 21 April 2018 (UTC)
replies: text:  Considering that the sinc function is less well-known than the ordinary sine function, no.  –Deacon Vorbis (carbon • videos) 01:23, 21 April 2018 (UTC)
replies: text: I don't find this argument convincing, because although sinc is indeed less well-known than sine, the whole purpose of Wikipedia is to be precise, and explain less well-known topics using text and links.  (It sounds like you were really just searching for a reason to justify the status quo.)  Not having heard any better arguments, I went ahead and made the change. Ross Finlayson (talk) 07:00, 24 April 2018 (UTC)
replies: text:  Wikipedia's purpose is not "to be precise".  There's a difference between being precise and being pedantic anyway, and the status quo was already correct and better.  That the product is given slightly rearranged doesn't change how it's generally referred to.  Please just drop this one.  –Deacon Vorbis (carbon • videos) 12:10, 24 April 2018 (UTC)
thread_title: "sine function", or "sinc function"?

message: Here is a suggested addition to the content, with reference.  I have requested similar text be added to the page Thomas J. Osler and Viète's formula.
In 1999, American mathematician Thomas J. Osler discovered that Viète's formula (1593) and the Wallis product are two special cases of a more general infinite product, which has been referred to as the Osler product by Arndt and Haenel, according to whom:
"The Osler product [which takes a parameter p] turns into the Viète product as p tends to infinity, and is equal to the Wallis product when p = 0.  In the intermediate cases p = 1, p = 2, etc., we obtain combined Viète- and Wallis-like products"
Will an appropriate editor please make this addition?  I have a COI.  Thank you! Skymath1 (talk) 04:04, 28 November 2020 (UTC)
replies: text: Suggest we consolidate any discussion at Talk:Viète's formula#request for mention of "Osler product". Russ Woodroofe (talk) 22:10, 29 November 2020 (UTC)
thread_title:  request for addition regarding "Osler Product"