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Math Help - Estimate oredr O(h)

  1. #1
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    Estimate oredr O(h)

    Consider non-dimensional differential equation for the height at the highest point is given by
    \begin{equation} h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} log_e(1+\mu) \end{equation}
    $0<\mu<<1.$
    Deduce an estimate to $O(\mu)$ for $h(\mu)$ and compare with $t_h(\mu)=1-\frac{\mu}{2}+...$
    => I really don't how to start this question. please help me.
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  2. #2
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    Re: Estimate oredr O(h)

    I would start with a Taylor's series expansion of ln(1+ \mu).
    Thanks from grandy
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  3. #3
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    Re: Estimate oredr O(h)

    Quote Originally Posted by HallsofIvy View Post
    I would start with a Taylor's series expansion of ln(1+ \mu).
    $\log_e(1+\mu) = \mu - \dfrac{\mu^2}{2} + \dfrac{\mu^3}{3} - \dfrac{\mu^4}{4}+\cdots$ and plug that in,
    to get $\log_e(1+\mu) =\dfrac12- \dfrac\mu3+\dfrac{\mu^2}{4}+\cdots.$
    now, comparing $\log_e(1+\mu)$ with $t_h(\mu)=1-\frac{\mu}{2}+\cdots$
    what can I say?
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  4. #4
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    Re: Estimate oredr O(h)

    Quote Originally Posted by grandy View Post
    Consider non-dimensional differential equation for the height at the highest point is given by
    \begin{equation} h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} log_e(1+\mu) \end{equation}
    $0<\mu<<1.$
    Deduce an estimate to $O(\mu)$ for $h(\mu)$ and compare with $t_h(\mu)=1-\frac{\mu}{2}+...$
    => I really don't how to start this question. please help me.
    well done................
    Soran University
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