Power Series Expansion for Hyperbolic Tangent Function

Theorem
The hyperbolic tangent function has a Taylor series expansion:

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

This converges for $\size x < \dfrac \pi 2$.

Proof
From Power Series Expansion for Hyperbolic Cotangent Function:
 * $(1): \quad \coth x = \ds \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}$

Then:

By Combination Theorem for Limits of Real 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 $\size x < \dfrac \pi 2$.