Find Taylor Series for 1/x Around x=3

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Homework Help Overview

The discussion revolves around finding the Taylor series for the function f(x) = 1/x centered at x = 3. Participants are exploring the correct formulation of the series and the derivatives involved in the process.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the derivatives of the function and their expressions, questioning the correctness of the Taylor series representation. There is uncertainty about the proper centering of the series and the coefficients involved.

Discussion Status

Some participants have pointed out potential errors in the formulation of the Taylor series and the derivatives. There is an ongoing examination of the general formula for Taylor series, with suggestions to verify the coefficients and their calculations.

Contextual Notes

There is a mention of the necessity to compute the derivatives at the center point (x = 3) rather than at a general x, indicating a specific requirement for the problem setup. Participants are also reflecting on the implications of dropping coefficients in their calculations.

soitgoes2019
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Homework Statement


Find the Taylor Series for f(x)=1/x about a center of 3.

Homework Equations

The Attempt at a Solution


f'(x)=-x^-2
f''(x)=2x^-3
f'''(x)=-6x^-4
f''''(x)=24x^-5
...
f^n(x)=-1^n * (x)^-(n+1) * (x-3)^n
I'm not sure where I went wrong...
 
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How did you write the sum, i.e. the requested Taylor series? And what is wrong or why do you think it is wrong?
 
I wrote the sum from n=0 to ∞ as: ∑-1^n (x)^-(n+1) (x-3)^n
I'm not sure if that is correctly centered at 3
 
As far as I can see, there is only a minor error; however crucial to the usage of Taylor series. You should check the general formula again.

Edit: And your formula for ##f^{(n)}## is wrong. You must not drop the coefficients all of a sudden, only because they might cancel out later in the calculation.
 
soitgoes2019 said:

Homework Statement


Find the Taylor Series for f(x)=1/x about a center of 3.

Homework Equations

The Attempt at a Solution


f'(x)=-x^-2
f''(x)=2x^-3
f'''(x)=-6x^-4
f''''(x)=24x^-5
...
f^n(x)=-1^n * (x)^-(n+1) * (x-3)^n
I'm not sure where I went wrong...

When you expand ##f(x)## about ##x = 3## your coefficients involve ##f^{(n)}(3)##, not ##f^{(n)}(x)##. But, of course, you compute ##f^{(n)}(3)## by first computing ##f^{(n)}(x)## and then setting ##x = 3##.
 

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