Power Series Expansion for Hyperbolic Cotangent Function

Theorem
The hyperbolic cotangent function has a Taylor series expansion:

where $B_{2 n}$ denotes the Bernoulli numbers.

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

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

This is less than $1$ :
 * $\left|{x}\right| < \pi$

Hence by the Ratio Test, the series converges for $\left|{x}\right| < \pi$.