Power Series Expansion for Cosecant Function

Theorem
The cosecant function has a Laurent series expansion:


 * $\displaystyle \csc x = \sum_{n \mathop = 0}^\infty \dfrac {\left({-1}\right)^{n-1} 2 \left({2^{2n-1}-1}\right) B_{2n}} {\left({2n}\right)!} x^{2n-1}$

where $B_n$ denotes the Bernoulli numbers.

This converges for $0 < \left|{x}\right| < \pi$.

Convergence
By Combination Theorem for Limits of Functions we can deduce the following.

This is less than $1$ if $\left|{x}\right| < \pi$.

Hence by the Ratio Test, the outer radius of convergence is $\pi$

The Principle Part of the laurent series is finite so converges for $x\neq0$. Thus we have the annulus of convergence to be $0<|x|<\pi$