Power Series Expansion for Real Area Hyperbolic Sine/Lemma 2

Lemma for Power Series Expansion for Real Area Hyperbolic Sine

 * $\map \ln {1 + \sqrt {1 + x^2} } = \dfrac 1 2 \cdot \dfrac {x^2} 2 - \dfrac {1 \times 3} {2 \times 4} \cdot \dfrac {x^4} 4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} \cdot \dfrac {x^6} 6 - \cdots$

This holds for $x \in \R_{\ne 0}$ such that $-1 < x < 1$.