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Math Help - Taylor polynomial...

  1. #1
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    Taylor polynomial...

    Find the n-th degree Taylor polynomial P_n(x) of f(x)=\ln x at c=1. Calculate P_3(1.1) and estimate the accuracy of the approximation \ln 1.1 P_3(1.1).

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  2. #2
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    The Taylor series expansion of f(x) about c = 1 is:
     f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-1)^n .
    This is also P_n(x). For  f(x) = ln(x) ,  f(1) = 0 and  f^{(n)}(1) = (-1)^{n+1} (n-1)! . So:
     ln(x) = \sum_{n = 1}^{\infty} \frac{(-1)^n}{n}(x-1)^n .
    To calculate P_3(1.1), let x = 1.1 and add up the terms in the series up to n = 3. You should get something close to ln(1.1).
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