Power Series Expansion for Tangent Function

Theorem
The tangent function has a Taylor series expansion:


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

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

This converges for $\left|{x}\right| < \dfrac \pi 2$.

Proof
First we will derive the Laurent series for $\cot x$.

Then:

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

This is less than $1$ if $\left|{x}\right| < \dfrac \pi 2$.

Hence by the Ratio Test, the series converges for $\left|{x}\right| < \dfrac \pi 2$.