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Math Help - Maclaurins series expansion and limits

  1. #1
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    Maclaurins series expansion and limits

    I cannot work out how i would start this question.

    Use the Maclaurin series to evaluate the limit
    lim x →0 ((x cos x − x)/sin^3x)

    Any help is much appreciated
    Thank you once again
    ViperRobK
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  2. #2
    Eater of Worlds
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    \lim_{x\to 0}\frac{xcos(x)-x}{sin^{3}(x)}

    Use the respective known series for sine and cosine.

    sin(x)=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k+1}}{(2k+1)!}

    xcos(x)=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k+1}}{(2k)!}

    The first few terms:

    \lim_{x\to 0}\left[\frac{\frac{-x^{3}}{2!}+\frac{x^{5}}{4!}-\frac{x^{7}}{6!}+\frac{x^{9}}{8!}-.................}{x^{3}-\frac{x^{5}}{2}+\frac{13x^{7}}{120}-\frac{41x^{9}}{3024}+............}\right]
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