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Infinite product convergence

  1. Jan 4, 2009 #1
    does the following infinite product converge ? and what to ?

    [tex]\prod^{\infty}_{n=1}(1-\frac{x^n}{n})[/tex]

    i know it has something to do with elliptic functions ....
     
  2. jcsd
  3. Jan 4, 2009 #2

    jambaugh

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    I'm not sure except I believe so for |x|<1.
    You can apply series test to the infinite sum you get by taking the logarithm of the infinite product.

    Consider also the power series expansion:
    [tex]\ln \left(1+z\right) = \sum_{k=1}^{\infty} (-)^{k+1}z^k[/tex]
    for [itex] |z|<1[/itex].
    So you can express the logarithm of your product as an infinite sum of power series.
    See if reordering gives you an answer you can use.
     
  4. Jan 7, 2009 #3
    Be careful with reordering! If a series like that only conditionally converges, then by reordering, the series can converge to a different value (or even diverge).

    If it helps, Mathematica is unable to simplify that product, except for trivial values of x.
     
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