Power Series Expansion for Real Area Hyperbolic Sine/Lemma 2

Lemma for Power Series Expansion for Real Area Hyperbolic Sine

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

This holds for $x \in \R: \size x < 1$.

Proof
It remains to determine the value of the arbitrary constant $C$.

Let $x \to 0$.

Then we have: