Results 1 to 3 of 3

Math Help - Wallis product.

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    4

    Wallis product.

    Wallis product.-arctan.jpg
    Last edited by mr fantastic; August 16th 2010 at 10:12 PM. Reason: Re-titled.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17
    The first part is straightforward, just plug in x = 1. For the second part, what is meant by compare? Are you asked to show which one converges faster per the number of terms?

    Btw, the Wallis product for pi is

    [Math]\displaystyle \prod_{n=1}^{\infty} \frac{2n}{2n-1} \cdot \frac{2n}{2n+1} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \frac{\pi}{2}[/tex]
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    The Leibnitz series is...

    \displaystyle \frac{\pi}{4} = (1-\frac{1}{3}) + (\frac{1}{5} - \frac{1}{7}) + ... + (\frac{1}{2n-1} - \frac{1}{2n+1}) + ... (1)

    ... so that after 2n iterations the last summed term is \frac{1}{2n-1} - \frac{1}{2n+1} = \frac{2}{4n^{2}-1}...

    The Wallis product [in logarithmic version...] is...

    \displaystyle \ln \frac{\pi}{2} =  \ln (1+\frac{1}{3}) + \ln (1 + \frac{1}{15}) + ... + \ln(1+\frac{1}{4n^{2}-1}) + ... (2)

    ... so that after 2n iterations the last summed term is \ln (1+ \frac{1}{4 n^{2} -1}) \sim \frac{1}{4 n^{2}-1} ...

    The two methods seem to be comparable in speed: both are very slow ...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Wallis' integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 8th 2010, 01:41 AM
  2. Easy question about Kruskal Wallis Test
    Posted in the Statistics Forum
    Replies: 0
    Last Post: August 18th 2010, 11:06 AM
  3. [SOLVED] Modified Wallis Example
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: June 7th 2010, 05:25 AM
  4. wallis product problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 16th 2008, 10:07 PM
  5. Replies: 1
    Last Post: May 14th 2008, 12:31 PM

Search Tags


/mathhelpforum @mathhelpforum