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

[mona123]

Let \(\displaystyle \nu\ge 1\) be a parameter.

For all \(\displaystyle t>0,\) we consider

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

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

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

Can you please help me to do so.

Thanks.