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Math Help - Error Term Taylor Formula

  1. #1
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    Question Error Term Taylor Formula

    Use the following form of the error term

     <br /> <br />
R_{n+1}(x) = \frac{f^{n+1}(c)}{(n+1)!} x^{n+1}<br />

    where c is between 0 and x, to determine in advance the degree of Taylor polynomial at a = 0 that would achieve the indicated accuracy in the interval [0,x] for...

    ...  <br /> <br />
f(x) = e^x, x = 2, error < 10^{-3}<br />

    (Do not compute Taylor polynomial.)

    Thanks for the help!
    Last edited by juicysharpie; February 25th 2009 at 02:01 AM.
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  2. #2
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    Quote Originally Posted by juicysharpie View Post
    Use the following form of the error term

     <br /> <br />
R_{n+1}(x) = \frac{f^{n+1}(c)}{(n+1)!} x^{n+1}<br />

    where c is between 0 and x, to determine in advance the degree of Taylor polynomial at a = 0 that would achieve the indicated accuracy in the interval [0,x] for...

    ...  <br /> <br />
f(x) = e^x, x = 2, error < 10^{-3}<br />

    (Do not compute Taylor polynomial.)

    Thanks for the help!
    Using your remainder formula with f(x) = e^x on [0,2]

     <br /> <br />
R_{n+1}(2) = \frac{e^c }{(n+1)!} 2^{n+1} \le \frac{e^2 }{(n+1)!} 2^{n+1} < \frac{9}{(n+1)!} 2^{n+1} \le 10^{-3}<br />
which gives n = 10 although nine terms will work (trial and error).
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