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

In summary: Lizwi discussed using the MacClaurin series for e^x to expand the function e(a+x)^2. Lizwi mentioned using the chain rule for the nth derivative and expanding the series about the point x=0 to simplify the terms. DonAntonio suggested using the MacClaurin series for e^x with the function (a+x)^2. The MacClaurin series for e^x is e^x=\sum_{n=0}^\infty\frac{x^n}{n!}\, and it converges for any real number. In summary, they discussed using the MacClaurin series for e^x to expand the function e(a+x)^2 by finding the nth derivative and simplifying
  • #1
Lizwi
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how do you do the taylor series expansion of e(a+x)2
 
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  • #2
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?
 
  • #3
Yes, I think you use the chain rule
 
  • #4
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 dissappear as a result of this?
 
  • #5
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
 

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

What is a Taylor series expansion?

A Taylor series expansion is a mathematical representation of a function as an infinite sum of terms, which are calculated from the values of the function and its derivatives at a single point.

Why is the Taylor series expansion important?

The Taylor series expansion allows us to approximate a function with a polynomial, making it easier to analyze and manipulate. It also allows us to evaluate functions at points where they are difficult or impossible to calculate directly.

What is the difference between a Taylor series and a Maclaurin series?

A Taylor series expansion is a representation of a function around a specific point, while a Maclaurin series is a special case of the Taylor series expansion, where the point of expansion is 0.

How do you find the coefficients for a Taylor series expansion?

The coefficients for a Taylor series expansion can be found by taking the derivatives of the function at the point of expansion and plugging them into the Taylor series formula.

What is the purpose of the remainder term in a Taylor series expansion?

The remainder term in a Taylor series expansion represents the difference between the exact value of the function and the approximation given by the Taylor series. It helps us to understand the accuracy of the approximation and to determine how many terms of the series are needed for a desired level of precision.

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