Power Series Expansion for Real Area Hyperbolic Sine/Mistake

Source Work

 * $8$. Taylor Series and Power Series:
 * Appendix: Table $8.2$: Power Series of Important Functions

Mistake

 * $\sinh^{-1} x = \paren {\ln 2} x + \dfrac 1 {\paren {2 \times 2} x^2} - \dfrac {1 \times 3} {\paren {2 \times 4 \times 4} x^4} + \dfrac {1 \times 3 \times 5} {\paren {2 \times 4 \times 6 \times 6} x^6} + \cdots \quad \paren {\size x \ge 1}$

Correction
There are two mistakes here:


 * $(1): \quad$ The first term should read $\map \ln {2 x}$


 * $(2): \quad$ The formula for $x \le -1$ should be:


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