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Math Help - infinite product convergence

  1. #1
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    infinite product convergence

    i'm having trouble trying to expand the following infinite product .

    \prod^{\infty}_{n=1}(1-\frac{x^n}{n})

    so , does it converge ? and what to ? i know it has something to do with elliptic functions ....but don't know where to start !
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by mmzaj View Post
    i'm having trouble trying to expand the following infinite product .

    \prod^{\infty}_{n=1}(1-\frac{x^n}{n})

    so , does it converge ? and what to ? i know it has something to do with elliptic functions ....but don't know where to start !
    Note that \prod_{n=1}^{\infty}\left\{1-\frac{x^n}{n}\right\}=\exp\left(\ln\prod_{n=1}^{\i  nfty}\left\{1-\frac{x^n}{n}\right\}\right) consider that we may rewrite \ln\prod_{n=1}^{\infty}\left\{1-\frac{x^n}{n}\right\} as \sum_{n=1}^{\infty}\ln\left(1-\frac{x^n}{n}\right). So obviously \prod_{n=1}^{\infty}\left\{1-\frac{x^n}{n}\right\}=\exp\left(\sum_{n=1}^{\infty  }\ln\left(1-\frac{x^n}{n}\right)\right) is only going to be finite if \sum_{n=1}^{\infty}\ln\left(1-\frac{x^n}{n}\right) is finite. For the value think the infinite product for \text{sinc}(x)=\frac{\sin(\pi x)}{\pi x}

    EDIT: Note that the last part is incorrect. I do not know where my head was this is not the since function
    Last edited by Mathstud28; January 5th 2009 at 01:23 PM.
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