Find Taylor Series for 1/x Around x=3

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SUMMARY

The discussion focuses on finding the Taylor Series for the function f(x) = 1/x centered at x = 3. The derivatives calculated include f'(x) = -x^-2, f''(x) = 2x^-3, f'''(x) = -6x^-4, and f''''(x) = 24x^-5. The correct Taylor series expansion involves evaluating the derivatives at the center point, specifically f^{(n)}(3), rather than f^{(n)}(x). The user initially misrepresented the series sum, which should be corrected to reflect the proper coefficients derived from the derivatives evaluated at x = 3.

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  • Understanding of Taylor Series and their applications
  • Knowledge of calculus, specifically differentiation
  • Familiarity with evaluating functions and their derivatives at specific points
  • Basic algebraic manipulation skills
NEXT STEPS
  • Review the derivation of Taylor Series for different functions
  • Learn how to compute higher-order derivatives effectively
  • Practice evaluating derivatives at specific points, such as f^{(n)}(3)
  • Explore the convergence of Taylor Series and their radius of convergence
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Students studying calculus, particularly those focusing on series expansions, as well as educators teaching Taylor Series concepts and applications.

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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