Power series expansion of an exponential

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Homework Help Overview

The discussion revolves around expanding the exponential term in the equation y=2[e^{x+(x²/2)}-1] as a power series. Participants are exploring the appropriate method for substitution and expansion within the context of power series and exponential functions.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss whether to substitute x+(x²/2) directly into the power series formula for e^x and how to handle the resulting series. There are questions about the implications of the substitution and the expected form of the final expression.

Discussion Status

The discussion is active, with participants providing suggestions and clarifications. Some guidance has been offered regarding the substitution process, but there is still uncertainty about the correctness of the resulting series and its alignment with a previously worked solution. Multiple interpretations of the problem are being explored.

Contextual Notes

There is mention of a previously worked solution that is expected to agree with the power series expansion only up to the first power of x, which raises questions about the requirements of the problem. Participants are also noting potential discrepancies in coefficients and the reasoning behind the problem's constraints.

t_n_p
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Homework Statement



expand the exponential term in the equation y=2[e^{x+(x²/2)}-1] as a power series

Homework Equations



on wikipedia I found this...
http://img297.imageshack.us/img297/1088/15139862vw6.jpg

The Attempt at a Solution


Do I substitute x+(x²/2) as "x" in the above formula and proceed as normal or must I do something different?
 
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t_n_p said:

Homework Statement



The Attempt at a Solution


Do I substitute x+(x²/2) as "x" in the above formula and proceed as normal or must I do something different?

Yes, just substitute.
 
Don't forget to multiply each term by 2 and subtract 1 from the constant term.
 
HallsofIvy said:
Don't forget to multiply each term by 2 and subtract 1 from the constant term.

And brush your teeth! :biggrin:
 
Thanks for the quick replies.
Using "x" = x+x²/2

I get y=2{1 + x + x²/2 + (x+x²/2)²/2 +...}-1
this leads to..

y=2x+2x²+(x^3)+(x^4)/4

I think I may have made a mistake but I cannot see where. My reason being I am supposed to show that a previously worked solution of y=2x+x²+c and the original equation y=2[e^{x+(x²/2)}-1] agree up to the first power of x only.

firstly, in the previously worked solution i am missing a coefficient of 2 for x². Secondly, why would the question ask to show that the original solution only agrees with the power series expansion of the same equation only to the power of x? It makes no sense!
 
I suggest you post the relevant question here. We can't figure out what's wrong unless we know what the question asks.
 

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