Power Series Expansion for Real Area Hyperbolic Tangent

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

for $\left\lvert{x}\right\rvert < 1$.

Proof
From Sum of Infinite Geometric Progression:
 * $(1): \quad \displaystyle \frac 1 {1 - x^2} = \sum_{n \mathop = 0}^\infty \left({x^2}\right)^n$

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 $\left\langle{\dfrac {x^{2 n + 1}} {2 n + 1} }\right\rangle$ is decreasing and converges to zero.

Hence the result.