Binomial expansion of a function with x raised to a power

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Discussion Overview

The discussion focuses on the binomial expansion of the function \(\frac{1}{(1-x^{2})}\) and its application to calculating the product \(\prod^{∞}_{i=1}\frac{1}{(1-x^{i})}\) for \(i\) up to 6. Participants explore the method of binomial expansion and its implications for further calculations.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant seeks clarification on how to binomial expand \(\frac{1}{(1-x^{2})}\) and believes that the expansion can be expressed in a specific series format.
  • Another participant provides a link to a resource that may assist with understanding binomial series.
  • A third participant confirms the initial participant's expansion approach and points out specific values in the expansion, suggesting a pattern may emerge.
  • A fourth participant expresses a desire to continue working on the problem without receiving further input at that moment.

Areas of Agreement / Disagreement

There is some agreement on the correctness of the proposed binomial expansion approach, but the discussion remains open with participants still exploring the topic and working through the details.

Contextual Notes

Participants have not fully resolved the mathematical steps involved in the expansion or the implications for the product calculation. There may be assumptions regarding the convergence of the series that are not explicitly stated.

Dixanadu
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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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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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