Power Series Expansion for Real Arccosecant Function

Theorem
The arccosecant function has a Taylor Series expansion:

which converges for $\size x \ge 1$.

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

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

which is converges for $\size x \le 1$.

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

This converges for $\size {\dfrac 1 x} \le 1$, that is, for $\size x \ge 1$.

Also see

 * Power Series Expansion for Real Arcsine Function


 * Power Series Expansion for Real Arccosine Function


 * Power Series Expansion for Real Arctangent Function


 * Power Series Expansion for Real Arccotangent Function


 * Power Series Expansion for Real Arcsecant Function