Solving Taylor Series: (1-x^2)^(-0.5) Help

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Homework Help Overview

The discussion revolves around finding the Taylor series for the function (1-x^2)^(-0.5) by first determining the series for (1-x)^(-0.5). Participants explore different methods to achieve this, including substitution and multiplication of series expansions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the initial expansion of (1-x)^(-0.5) and consider how to apply it to (1-x^2)^(-0.5). There is mention of substituting x^2 into the expansion and questioning whether multiplying the expansions would yield the same result. Some express uncertainty about the complexity of multiplying the two series.

Discussion Status

The discussion is active, with participants sharing insights about substitution and the potential challenges of multiplying series. There is no explicit consensus on the best approach, but several lines of reasoning are being explored.

Contextual Notes

Participants note that the problem is a one-mark question, which raises questions about the expected simplicity of the solution. There is also mention of the tedious nature of determining coefficients when multiplying series.

dan38
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Homework Statement


Have to find the Taylor series for (1-x)^(-0.5)
Then use this to find the Taylor series for (1-x^2)^(-0.5)

Homework Equations





The Attempt at a Solution


Was able to do the expansion for the first one quite easily, but not sure how to do the second one. My initial thought was to multiply the first expansion with the expansion for (1+x)^(-0.5)
since (1-x^2) = (1-x)(1+x)
But this is a one mark question, so this seems like too much working out to be the right way...
Is there a trick to this question? :S
 
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dan38 said:

Homework Statement


Have to find the Taylor series for (1-x)^(-0.5)
Then use this to find the Taylor series for (1-x^2)^(-0.5)

Homework Equations



The Attempt at a Solution


Was able to do the expansion for the first one quite easily, but not sure how to do the second one. My initial thought was to multiply the first expansion with the expansion for (1+x)^(-0.5)
since (1-x^2) = (1-x)(1+x)
But this is a one mark question, so this seems like too much working out to be the right way...
Is there a trick to this question? :S
If [itex]\displaystyle f(x)=\frac{1}{\sqrt{1-x}}\,,[/itex] then what is [itex]f(x^2)\ ?[/itex]
 
ah yes that's it, so I would just sub in x^2 to the expansion
thanks!
Would multiplying the two expansions have also worked?
 
dan38 said:
ah yes that's it, so I would just sub in x^2 to the expansion
thanks!
Would multiplying the two expansions have also worked?
Multiplying which two expansions ?
 
dan38 said:
ah yes that's it, so I would just sub in x^2 to the expansion
thanks!
Would multiplying the two expansions have also worked?

Much, much more troublesome. You have to figure out which terms contribute to the coefficient of each term in the final expansion. For example, the [itex]x^3[/itex] term in the product is determined by the sum of the [itex]x^3[/itex] terms of both expressions, as well as the sum of the products of the [itex]x[/itex] term of one with the [itex]x^2[/itex] term of the other (and vice versa). The general term would involve lots of tedious algebra.

It wouldn't be too bad if they just asked you to find the expression correct to the first few terms, though.
 
No, squaring the original series would give [itex]\left(1/\sqrt{1- x}\right)^2= 1/\sqrt{1- x}[/itex], not [itex]1/(1- x^2)[/itex].
 

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