Is there a power series for $\text{sech}(x)=\frac{2}{e^{-x}+e^{x}}$ that converges for all $x$ or for $x>\frac{\pi}{2}$? I am familiar with the well known series \begin{equation} \text{sech}(x)=\sum_{n=0}^{\infty}\frac{E_{2n}}{(2n)!}x^{2n}, \end{equation} where the $E_{2n}$ are the Euler numbers. This series converges for $x^2<\frac{\pi^2}{4}$. It is interesting that this series stops working at $\frac{\pi}{2}$ considering the power series for $e^x$ converges for all $x$, but it definitely does.
So is it possible to expand $\text{sech}(x)$ in powers of $x$ beyond $\frac{\pi}{2}$?
There is no globally convergent power series for sech(x), but you can of course get the taylor series for any specific point.
However there are certain series' of that do converge globally that can be computed efficiently:
Consider that
$$sech(x) = \frac{1}{cosh(x)} = \frac{2}{e^x+e^-x}$$
We can now use the taylor series of cosh(x):
$$sech(x) = \frac{1}{cosh(x)} = \frac{1}{\sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}}$$
This series also converges extremely quickly.