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Thread: Taylor's series: A(t) = [1-cos(t*sqrt{4v-1})]/[4v-1] - (cosh(t)-1), g(t) = ...

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    Taylor's series: A(t) = [1-cos(t*sqrt{4v-1})]/[4v-1] - (cosh(t)-1), g(t) = ...

    Let [tex]\nu\ge 1[/tex] be a parameter.

    For all [tex]t>0,[/tex] we consider

    . . . . .[tex]\begin{align}
    A(t) & =\frac{1-\cos(t\sqrt{4\nu-1})}{4\nu-1}-(\cosh(t)-1) \\[10pt]
    g(t) & =\frac{\frac{\sin(t\sqrt{4\nu-1})}{\sqrt{4\nu-1}}+\sinh(t)}{A(t)}
    \end{align}[/tex]

    By using Taylor's series, I want to prove that there exists a constant [tex]c>0[/tex] which doesn't depend on [tex]\nu[/tex] such that

    . . . . .[tex]\dfrac{\nu t^3}{\ln\left(1-\frac{2}{g(t)+1}\right)}\le c\, \mbox{ for all }\, t\le \dfrac{1}{\sqrt{4\nu-1}}[/tex]

    Can you please help me to do so.

    Thanks.
    Last edited by stapel; 12-15-2017 at 04:00 PM. Reason: Tweaking LaTeX.

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