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Math Help - Taylor Theorum

  1. #1
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    Taylor Theorum

    In a Taylor polynomial, how would I find the upper bound for the error of the approximation: arctan(0.4) = 0.4 - (0.4)^3/3! . Thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by me > jallison + brownie
    In a Taylor polynomial, how would I find the upper bound for the error of the approximation: arctan(0.4) = 0.4 - (0.4)^3/3! . Thanks
    Look at the form of the remainder of the truncated Taylor series. This is an
    alternating series (the sign of consecutive terms is alternately + and -) of
    decreasing terms so the absolute error is less than the absolute value first
    ignored term. In this case the Taylor series is (as we are interested in
    |x|<1):

    <br />
\arctan(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-...<br />

    (note you have an error in your Taylor polynomial).

    So in this case if we truncate after the second term the absolute error in the
    approximation is less than:

    <br />
\left| \frac{x^5}{5} \right|=\frac{(0.4)^5}{5}\approx 0.002<br />

    RonL
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