Power Series Expansion for Cotangent Function

Theorem
The (real) cotangent function has a Taylor series expansion:

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

This converges for $0 < \size x < \pi$.

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

This is less than $1$ :
 * $\size x < \pi$

Hence by the Ratio Test, the series converges for $\size x < \pi$.

Also see

 * Laurent Series Expansion for Cotangent Function