$\sech z$ has an even Taylor series whose radius $\pi/2$ is set by the nearest zeros of $\cosh z$.Suggest a slogan
For \(z\in\CC\) with \(\abs{z}<\pi/2\), \[ \sech z = \frac{1}{\cosh z} = 1-\frac{z^{2}}{2}+\frac{5 z^{4}}{24}-\frac{61 z^{6}}{720}+\cdots. \] The radius of convergence is \(\pi/2\), the distance from \(0\) to the nearest zeros \(\pm i\pi/2\) of \(\cosh\).
What is the series expansion of $\sech(z)$?