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Math Help - How big should n be?

  1. #1
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    How big should n be?

    Let p(x) be the taylor polynomial of degree n of the function
    f(x) = log(1-x), about a = 0. How large should n be chosen to have
    |f(x) - p(x)| <= 10^-4 ?

    So I know how to go about this, but for Rn, what is f^(n+1) ?
    Which derivative do I use? How do I know which to do?

    Can someone give me the approximate n I'm looking for and the formula you used to get there, with details?
    Thanks
    Last edited by amor_vincit_omnia; August 22nd 2008 at 08:51 AM.
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  2. #2
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    You probably should generate the series, first. My guess is that it is easier than you might think.
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  3. #3
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    Quote Originally Posted by TKHunny View Post
    You probably should generate the series, first. My guess is that it is easier than you might think.
    I'm sure I'm making it much harder.

    f(x) = log(1-x), f(a)= 0

    f'(x) = 1/(x-1), f'(a) = -1

    f''(x) = 1/(x-1)^2, f''(a) = -1

    So I'm thinking for f^(n+1) = -n! / (1-x)^(n+1)

    is this right?

    If so, |R(x)| <= |1/2|^(n+1)/ (n+1)! * (n!)/(1/2)^n+1

    = 1 / n+1

    right?
    So n would need to be what, like 5000?
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  4. #4
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    Quote Originally Posted by amor_vincit_omnia View Post
    I'm sure I'm making it much harder.

    f(x) = log(1-x), f(a)= 0

    f'(x) = 1/(x-1), f'(a) = -1
    Ack!! You should have stopped right there with the derivatives.

    \frac{1}{x-1}\;=\;-\frac{1}{1-x}\;=\;-\left(1+x+x^{2}+x^{3}+...\right)

    This leads immediately to:

    log(1-x)\;=\;-\left(x+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\frac{x^{4  }}{4}+...\right)

    This is NOT an alternating series. Are you SURE considering just one discarded term is sufficient?
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  5. #5
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    Confused. Clearly my teacher didn't go over this well, or I didn't grasp it well.
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  6. #6
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    The anti-derivative thing and uniform convergence probably would not have been covered in an introductory course. Your series of derivative was fine. Consider that a hint that once you ahve paid your dues there will be cooler ways to go about it.

    As far as bounds on the error, there must have been some discussion on the difference between alternating and non-alternating series. They ARE different animals. The result may be very similar, but getting there might take a different path.
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