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    Taylor polynomial

    Use taylor polynomial approximations to avoid loss of significance error.

    (x - sin(x)) / x^3

    Any advice? Thanks
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by jzellt View Post
    Use taylor polynomial approximations to avoid loss of significance error.

    (x - sin(x)) / x^3

    Any advice? Thanks
    Note that there is loss of significance at x=0.

    Thus, use the taylor series @ x=0 for sin x:

    \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots.

    So it turns out that

    \begin{aligned}\frac{x-\sin x}{x^3}&=\frac{x-\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots\right)}{x^3}\\ &=\frac{1}{3!}-\frac{x^2}{5!}+\frac{x^4}{7!}-\dots\end{aligned}

    Does this make sense?
    Last edited by Chris L T521; September 20th 2009 at 07:07 PM. Reason: made mistake.
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