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

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

Let $\nu\ge 1$ be a parameter.

For all $t>0,$ we consider

. . . . .\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}

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

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

Thanks.

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