Power Series Expansion for Real Area Hyperbolic Tangent

Theorem
The (real) area hyperbolic tangent function has a Taylor series expansion:

for $\size x < 1$.

Proof
From Sum of Infinite Geometric Sequence:
 * $(1): \quad \ds \frac 1 {1 - x^2} = \sum_{n \mathop = 0}^\infty \paren {x^2}^n$

for $-1 < x < 1$.

From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:

For $-1 < x < 1$, the sequence $\sequence {\dfrac {x^{2 n + 1} } {2 n + 1} }$ is decreasing and converges to zero.

Hence the result.

Also see

 * Power Series Expansion for Real Area Hyperbolic Sine
 * Power Series Expansion for Real Area Hyperbolic Cosine
 * Power Series Expansion for Real Area Hyperbolic Cotangent
 * Power Series Expansion for Real Area Hyperbolic Secant
 * Power Series Expansion for Real Area Hyperbolic Cosecant