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Infinite product

  1. Oct 21, 2007 #1
    When 0 < x < 1, we can be sure that the following infinite product converges.

    [tex]\[ \prod_{n=1}^\infty (1 + x^n)\][/tex]

    But is there a closed form for it?
     
  2. jcsd
  3. Oct 21, 2007 #2
    Not really an answer to your question, but the expression you've post is the way of generating the partitions of integers.

    Write out the first few terms, [tex](1+x)(1+x^{2})(1+x^{3})(1+x^{4})(1+x^{5})(1+x^{6})(1+x^{7})[/tex]

    You can see that the coefficent of the constant term in the expansion will be just 1.
    Coefficent of the x term will be 1
    Coefficent of the x^2 term will be 1
    Coefficent of the x^3 term will be 2, because x^3 can be made from [itex]x.x^{2}[/itex] or just x^3 (ie 3 = 3+0 or 2+1)
    x^4 will have coefficent 2 because x^4 comes from x^4 and x^3 * x (ie 4 = 4+0 or 3+1)
    x^5 is 3, because 5 = 5+0 or 5=4+1 or 5=3+2
    x^6 is 4 because 6=6+0 or 6=5+1 or 6=4+2 or 6 = 3+2+1)

    See how it works? The coefficent of x^n will be the number of ways you can express n as a sum of unique integers. The closed form for this I don't think exists.
     
  4. Oct 21, 2007 #3
    Well, this is interesting. :smile: It appears that I am looking for the generating function of a very particular sequence, although I hadn't realize that. So finding a closed form seems against the odds...
     
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