Power Series Expansion for Real Area Hyperbolic Cosine/Lemma 1
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Lemma for Power Series Expansion for Real Area Hyperbolic Cosine
| \(\ds \dfrac 1 {\sqrt {1 - x^2} }\) | \(=\) | \(\ds 1 + \frac 1 2 x^2 + \frac {1 \times 3} {2 \times 4} x^4 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^6 + \cdots\) | ||||||||||||
| \(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} x^{2 n}\) |
for $x \in \R: -1 < x < 1$.
Proof
| \(\ds \dfrac 1 {\sqrt {1 - x} }\) | \(=\) | \(\ds 1 + \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^3 + \cdots\) | Power Series Expansion for $\dfrac 1 {\sqrt {1 - x} }$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac 1 {\sqrt {1 - x^2} }\) | \(=\) | \(\ds 1 + \frac 1 2 x^2 + \frac {1 \times 3} {2 \times 4} x^4 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^6 + \cdots\) | setting $x \gets x^2$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} x^{2 n}\) |
for $-1 < x < 1$.
$\blacksquare$