Binomial expansion of a function with x raised to a power

  1. Dixanadu

    Dixanadu 246
    Gold Member

    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. Office_Shredder

    Office_Shredder 4,500
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  4. HallsofIvy

    HallsofIvy 40,299
    Staff Emeritus
    Science Advisor

    Wait, wait- don't tell me. I'm still working on it!
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?

0
Draft saved Draft deleted
Similar discussions for: Binomial expansion of a function with x raised to a power
Loading...