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Math Help - approximating error on taylor poly

  1. #1
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    approximating error on taylor poly

    Find the fifth order taylor polynomial for f(x) = 1/x^2 based at x_0 = 1, then find an interval centered at x_0 = 1 in which the approximation error |f(x) - P_5(x)| < 0.01

    I found the taylor poly
    1 -2(x-1) + 3(x-1)^2 -4(x-1)^3 ... -6(x-1)^5

    I don't know how to simply find the approximation. I could try solving for x but that would take way too long.
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  2. #2
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    Quote Originally Posted by diroga View Post
    I found the taylor poly
    1 -2(x-1) + 3(x-1)^2 -4(x-1)^3 ... -6(x-1)^5

    I don't know how to simply find the approximation. I could try solving for x but that would take way too long.
    The error in an alternating series is always less than the n+1 term

    7(x-1)^6<\frac{1}{100} \iff (x-1)^6<\frac{1}{700}

    (x-1)< \left( \frac{1}{700} \right)^{\frac{1}{6}} \approx .336
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  3. #3
    Junior Member Infophile's Avatar
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    Hello,

    You have an expression of the rest : R(x)=\frac{1}{5!}\int_{1}^{x}f^{(6)}(t)(x-t)^5dt

    After having calculate this integral we obtain :

    R(x)=\frac{7(x^6-6x^5+15x^4-20x^3+15x^2-6x+1)}{x^8}

    Then you study this function on [0,2] for example.

    And we find this interval : [0,76;1,24]

    Maybe it exists an easier solution.

    Edit : Yes, I didn't think to use series...

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