Power Series Expansion for Cosecant Function

Theorem
The cosecant function has a Laurent series expansion:


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

where $B_n$ denotes the Bernoulli numbers.

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

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 principal part of the Laurent series is finite so converges for $x \ne 0$.

Thus we have the annulus of convergence to be $0 < \left\lvert{x}\right\rvert < \pi$.