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Taylor Series

  1. Jul 21, 2016 #1
    1. The problem statement, all variables and given/known data
    Find the Taylor Series for f(x)=1/x about a center of 3.

    2. Relevant equations


    3. 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...
     
  2. jcsd
  3. Jul 21, 2016 #2

    fresh_42

    Staff: Mentor

    How did you write the sum, i.e. the requested Taylor series? And what is wrong or why do you think it is wrong?
     
  4. Jul 21, 2016 #3
    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
     
  5. Jul 21, 2016 #4

    fresh_42

    Staff: Mentor

    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.
     
  6. Jul 21, 2016 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    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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