Power Series Expansion for Tangent Function/Proof of Convergence

Theorem
The radius of convergence of the Power Series Expansion for Tangent Function:
 * $\displaystyle \tan x = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}$

where $B_{2 n}$ denotes the Bernoulli numbers, is given as:
 * $\size x < \dfrac \pi 2$

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

This is less than $1$ :
 * $\size x < \dfrac \pi 2$

Hence by the Ratio Test, the series converges for $\size x < \dfrac \pi 2$.