I need to prove the Maclarin Expansion of $\displaystyle Sinh (x)$.I know $\displaystyle Sinh (x) = (e^x - e^-x)/2$ but where do go from here? Please Help.
Hello rideorsmurfUse the expansions of $\displaystyle e^x$ and $\displaystyle e^{-x}$:
$\displaystyle e^x = 1 + x + \frac{x^2}{2!}+ \frac{x^3}{3!}+ \frac{x^4}{4!}+ \frac{x^5}{5!}+ \dots$
$\displaystyle e^{-x} = 1 - x + \frac{x^2}{2!}- \frac{x^3}{3!}+ \frac{x^4}{4!}- \frac{x^5}{5!}+ \dots$
... and subtract:
$\displaystyle e^x-e^{-x} = 2x +2\frac{x^3}{3!}+ 2\frac{x^5}{5!}+ \dots$
$\displaystyle \Rightarrow \sinh x =x +\frac{x^3}{3!}+ \frac{x^5}{5!}+ \dots$
Grandad