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Math Help - Maclaurin polynomial

  1. #1
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    Maclaurin polynomial

    Consider the Maclaurin polynomial of degree N for f(x)=e^x.

    i---Give an estimate for the error term when we use this to approximate
    e^(-1)

    ii---For what N will the Maclaurin polynomial at -1 give an approximationg for e^(-1) accurate to 6 decimal places. Compute this approximation.

    Well, I know the polynomial is 1+x+x^2/2+x^3/3!+...+x^N/N!

    For part i, do you just do 1/e-(1+(-1)+1/2-1/3+......+(-1)^N/N!)
    Is this the estimate of the error term? I'm not sure how you do this or part 2?
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  2. #2
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    Okay, for ii, I found n=9 just by trial and error, but how could I have gotten this mathematically?
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  3. #3
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    There is a formula, which I suspect you were expected to know, that says the error, in cutting off a Maclaurin series of a function at the nth power, is less than or equal to \frac{M}{n!}|x|^n where "M" is an upper bound on the n+1 derivative of the function between 0 and x.

    Here, since every derivative is again e^{x} and that has a maximum of 1 on any interval [-x, 0], that maximum error is \frac{1}{n!}< .000001. That is the same as saying that n!> 1000000
    8!= 40320< 1000000 and 9!= 362880> 1000000 so you need 9 terms.
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