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

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SUMMARY

The Taylor series expansion of the function e^(a+x)^2 can be performed by differentiating the function and evaluating its derivatives at x=0. The key steps involve using the chain rule to find the nth derivative and substituting x=0 to identify which terms cancel out. The Maclaurin series for e^x, expressed as e^x = Σ (x^n/n!), serves as a foundational tool for this expansion, as it converges for any real number.

PREREQUISITES
  • Understanding of Taylor series and Maclaurin series
  • Proficiency in differentiation techniques, particularly the chain rule
  • Familiarity with infinite series and convergence concepts
  • Basic knowledge of exponential functions and their properties
NEXT STEPS
  • Study the derivation and application of the Maclaurin series for e^x
  • Practice finding nth derivatives of composite functions using the chain rule
  • Explore convergence criteria for infinite series in calculus
  • Learn about applications of Taylor series in approximating functions
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus, as well as educators looking for effective methods to teach Taylor series expansions.

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

DonAntonio
 

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