Binomial expansion of a function with x raised to a power

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SUMMARY

The discussion focuses on the binomial expansion of the function \(\frac{1}{(1-x^{2})}\). Participants confirm that the expansion can be expressed as a series involving factorials and powers of \(-x^{2}\). The correct expansion is identified as \(1 + (-1)(-x^{2}) + \frac{(-1)(-2)(-x^{2})^{2}}{2!} + \frac{(-1)(-2)(-3)(-x^{2})^{3}}{3!} + \ldots\). Understanding this expansion is crucial for calculating the product \(\prod^{∞}_{i=1}\frac{1}{(1-x^{i})}\) for \(i\) up to 6.

PREREQUISITES
  • Understanding of binomial series and expansions
  • Familiarity with factorial notation and operations
  • Basic knowledge of infinite products in mathematics
  • Proficiency in manipulating algebraic expressions
NEXT STEPS
  • Study the binomial series and its special cases in detail
  • Learn how to compute infinite products, specifically \(\prod^{∞}_{i=1}\frac{1}{(1-x^{i})}\)
  • Explore the convergence criteria for binomial expansions
  • Practice binomial expansions with different powers and coefficients
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Mathematicians, students studying calculus or algebra, and anyone interested in series expansions and infinite products.

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

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