Power Series Expansion for Real Area Hyperbolic Sine/Lemma 1

Lemma for Power Series Expansion for Real Area Hyperbolic Sine
for $x \in \R: -1 < x < 1$.

Proof
From Power Series Expansion for $\dfrac 1 {\sqrt {1 + x} }$:

for $-1 < x < 1$.

The result follows on setting $x \gets x^2$.