Category:Power Series Expansion for Real Area Hyperbolic Sine

\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1} & : \size x < 1 \\ \ds \ln 2 x + \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } & : x \ge 1 \\ \ds -\ln \paren {-2 x} - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n \paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } } & : x \le -1 \\ \end {cases}\)

x - \dfrac {x^3} {2 \times 3} + \dfrac {1 \times 3 x^5} {2 \times 4 \times 5} - \dfrac {1 \times 3 \times 5 x^7} {2 \times 4 \times 6 \times 7} + \cdots & : \size x < 1 \\ \ln 2 x + \dfrac 1 {2 \times 2 x^2} - \dfrac {1 \times 3} {2 \times 4 \times 4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6 x^6} + \cdots & : x \ge 1 \\ -\ln \paren {-2 x} - \paren {\dfrac 1 {2 \times 2 x^2} - \dfrac {1 \times 3} {2 \times 4 \times 4 x^4} + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6 x^6} + \cdots} & : x \le -1 \\ \end {cases}\)