How Do You Perform the Taylor Series Expansion of e(a+x)2?

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Discussion Overview

The discussion revolves around performing the Taylor series expansion of the function \( e^{(a+x)^2} \). Participants explore the differentiation process required for the nth derivative and the application of the Maclaurin series.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant asks how to perform the Taylor series expansion of \( e^{(a+x)^2} \).
  • Another participant notes that the function is differentiable and continuous, suggesting it can be expanded at any real number point.
  • A participant mentions the use of the chain rule for differentiation.
  • It is suggested to expand the series about the point \( x=0 \) and to find the first few derivatives at that point.
  • Another participant introduces the Maclaurin series for \( e^x \) and suggests substituting \( (a+x)^2 \) into that series.

Areas of Agreement / Disagreement

Participants generally agree on the approach to differentiate the function and the applicability of the Maclaurin series, but there is no consensus on the specific steps or simplifications involved in the expansion process.

Contextual Notes

Participants have not fully resolved the mathematical steps involved in finding the nth derivative or the implications of substituting into the Maclaurin series.

Lizwi
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how do you do the taylor series expansion of e(a+x)2
 
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Lizwi said:
how do you do the taylor series expansion of e(a+x)2

Hey Lizwi and welcome to the forums.

This function is differentiable, continuous and can be expanded at any point where x is a real number.

To start off though, do you know how to differentiate your function for the nth derivative?
 
Yes, I think you use the chain rule
 
Lizwi said:
Yes, I think you use the chain rule

Yes you will do, but the key will be to expand your series about the point x=0 (in other words you need to find f'(0), f''(0) and so on).

If you end up getting a specific form (which you will) and then the zero's cancel out terms, then you will get a simplification for the nth derivative.

So expand out the first two or three derivatives (using the chain rule) and substitute in x = 0. What terms disappear as a result of this?
 
Lizwi said:
how do you do the taylor series expansion of e(a+x)2



Do you know the MacClaurin series for [itex]\,e^x\,[/itex] : [tex]e^x=\sum_{n=0}^\infty\frac{x^n}{n!}\,?[/tex] Well, you can now input [itex]\,(a+x)^2\,[/itex] above as this series converges for any real number...

DonAntonio
 

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