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Binomial expansion of a function with x raised to a power

by Dixanadu
Tags: binomial, expansion, function, power, raised
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Dixanadu
#1
Oct6-13, 12:26 PM
P: 123
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!
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RobinSky
#2
Oct6-13, 02:45 PM
P: 112
Maybe this will help:

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

Good luck!
Office_Shredder
#3
Oct6-13, 03:51 PM
Emeritus
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P: 4,500
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
#4
Oct7-13, 08:57 PM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,565
Binomial expansion of a function with x raised to a power

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


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