Power Series Expansion for Real Area Hyperbolic Secant

Theorem
The (real) area hyperbolic secant function has a Taylor series expansion:

for $0 < x \le 1$.

Proof
From Power Series Expansion for Real Area Hyperbolic Cosine:

for $x \ge 1$.

From Real Area Hyperbolic Cosine of Reciprocal equals Real Area Hyperbolic Secant:


 * $\map \arcosh {\dfrac 1 x} = \arsech x$

So:

For $1 \le \dfrac 1 x$ we have that $0 < x \le 1$.

Hence the result.

Also see

 * Power Series Expansion for Real Area Hyperbolic Sine
 * Power Series Expansion for Real Area Hyperbolic Cosine
 * Power Series Expansion for Real Area Hyperbolic Tangent
 * Power Series Expansion for Real Area Hyperbolic Cotangent
 * Power Series Expansion for Real Area Hyperbolic Cosecant