Binomial expansion of a function with x raised to a power

[Dixanadu]

Hey guys.

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

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

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

1+(-1)(-x^{2})+(-1)(-2)\frac{(-x^{2})^{2}}{2!}+(-1)(-2)(-3)\frac{(-x^{2})^{3}}{3!}+...

Is that correct?

Thanks guys!

 

[RobinSky]

Maybe this will help:

http://en.wikipedia.org/wiki/Binomial_series#Special_cases

Good luck!

 

[Office_Shredder]

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.

 

[HallsofIvy]

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