Power Series Expansion for Real Arccosecant Function

Theorem
The arccosecant function has a Taylor Series expansion:

which converges for $\left\lvert{x}\right\rvert \ge 1$.

Proof
From Arcsine of Reciprocal equals Arccosecant:
 * $\operatorname {arccsc} x = \arcsin \dfrac 1 x$

From Power Series Expansion for Real Arcsine Function:
 * $\displaystyle \arcsin x = \sum_{n \mathop = 0}^\infty \frac {\left({2 n}\right)!} {2^{2 n} \left({n!}\right)^2} \frac {x^{2 n + 1} } {2 n + 1}$

which is converges for $\left\lvert{x}\right\rvert \le 1$.

The result follows by subtituting $\dfrac 1 x$ for $x$.

This converges for $\left\lvert{\dfrac 1 x}\right\rvert \le 1$, that is, for $\left\lvert{x}\right\rvert \ge 1$