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Binomial expansion of a function with x raised to a power

  1. Oct 6, 2013 #1
    Hey guys.

    So I need to know how to Binomial expand the following function
    [itex]\frac{1}{(1-x^{2})}[/itex].

    I need this because I have to work out [itex]\prod^{∞}_{i=1}[/itex][itex]\frac{1}{(1-x^{i})}[/itex] for i up to 6. But I have to do it with Binomial expansion. If i can learn how to do [itex]\frac{1}{(1-x^{2})}[/itex] then the rest of the powers should be the same.

    I was under the impression that [itex]\frac{1}{(1-x^{2})}[/itex] can be binomial expanded as

    [itex]1+(-1)(-x^{2})+(-1)(-2)\frac{(-x^{2})^{2}}{2!}+(-1)(-2)(-3)\frac{(-x^{2})^{3}}{3!}+...[/itex]

    Is that correct?

    Thanks guys!
     
  2. jcsd
  3. Oct 6, 2013 #2
  4. Oct 6, 2013 #3

    Office_Shredder

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    This is correct, and you probably want to observe that
    (-1)/1! = -1
    (-1)(-2)/2! = 1
    (-1)(-2)(-3)/3! = -1

    and you can probably guess the pattern as you continue.
     
  5. Oct 7, 2013 #4

    HallsofIvy

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    Wait, wait- don't tell me. I'm still working on it!
     
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