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Math Help - MacLaurin series and AST error approximation

  1. #1
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    MacLaurin series and AST error approximation

    I gotta show that

    \frac{1}{2!}-\frac{x^2}{4!} < \frac{1-cos(x)}{x^2} < \frac{1}{2!} for x not equal to 0

    I know that |S-S_{n}| < a_{n+1} and S_{n-1} < S < S_{n} or S_{n} < S < S{n-1} depending on which one is bigger value and smaller value

    Using the MacLaurin series for cos x and applying \frac{1-cos (x)}{x^2} to it, I got

    \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n+1)!} = \frac{1}{2!} - \frac{x^2}{4!} + \frac{x^4}{6!}.....

    so clearly

    \frac{1}{2!} - \frac{x^2}{4!} < \,or\, > \frac{1 - cos(x)}{x^2} < \,or\, > \frac{1}{2!}

    what's the best way to show which of the two it is?


    \frac{1}{2!} -  \frac{x^2}{4!} < \frac{1}{2!} for all values of x except x=0

    ^^ just use this argument?
    Last edited by Jonroberts74; April 27th 2014 at 03:01 PM.
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  2. #2
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    Re: MacLaurin series and AST error approximation

    The error in an alternating series that has been truncated to n terms is never any more than the absolute value of the next term.

    So here if your series $\displaystyle \begin{align*} \frac{1 - \cos{(x)}}{x^2} = \frac{1}{2!} - \frac{x^2}{4!} + \frac{x^4}{6!} - \dots \end{align*}$ is truncated to two terms, then $\displaystyle \begin{align*} \frac{1 - \cos{(x)}}{x^2} \approx \frac{1}{2!} - \frac{x^2}{4!} \end{align*}$ with an error no greater than $\displaystyle \begin{align*} \frac{x^4}{6!} \end{align*}$.

    So clearly $\displaystyle \begin{align*} \frac{1}{2!} - \frac{x^2}{4!} < \frac{1 - \cos{(x)}}{x^2} \end{align*}$, since the next term along is being added, thereby showing that your estimate is an UNDER estimate.

    As for the other side of your inequality $\displaystyle \begin{align*} \frac{1}{2!} - \frac{x^2}{4!} < \frac{1 - \cos{(x)}}{x^2} < \frac{1}{2!} \end{align*}$, so where you are trying to show $\displaystyle \begin{align*} \frac{1 - \cos{(x)}}{x^2} < \frac{1}{2!} \end{align*}$ it is EXACTLY the same thing, just with your series truncated to one term. What would be the error then? Would the truncated series be an under estimate or an over estimate?
    Thanks from Jonroberts74
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