Power Series Expansion for Cosecant Function

Theorem
The cosecant function has a Taylor series expansion:


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

where $B_n$ denotes the Bernoulli numbers.

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

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