Power Series Expansion for Hyperbolic Secant Function

Theorem
The hyperbolic secant function has a Taylor series expansion:

where $E_{2 n}$ denotes the Euler numbers.

This converges for $\left|{x}\right| < \dfrac \pi 2$.

Proof
By definition of the Euler numbers:


 * $\displaystyle \operatorname {sech} x = \sum_{n \mathop = 1}^\infty \frac {E_n x^n} {n!}$

From Odd Euler Numbers Vanish:
 * $E_{2 k + 1} = 0$

for $k \in \Z$.

Hence the result.