Power Series Expansion for Real Area Hyperbolic Sine/Lemma 1

Lemma for Power Series Expansion for Real Area Hyperbolic Sine

$\ds \dfrac 1 {\sqrt {x^2 + 1} }$

$=$

$\ds 1 - \frac 1 2 x^2 + \frac {1 \times 3} {2 \times 4} x^4 - \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^6 + \cdots$

$\text {(1)}: \quad$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} x^{2 n}$

for $x \in \R: -1 < x < 1$.

$\ds \dfrac 1 {\sqrt {x + 1} }$

$=$

$\ds 1 - \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 - \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^3 + \cdots$

$\ds \leadsto \ \ $

$\ds \dfrac 1 {\sqrt {x^2 + 1} }$

$=$

$\ds 1 - \frac 1 2 x^2 + \frac {1 \times 3} {2 \times 4} x^4 - \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^6 + \cdots$

setting $x \gets x^2$

$\ds $

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} x^{2 n}$

for $-1 < x < 1$.

$\blacksquare$