Hi, how do I find the Taylor's series around zero for this:
f(x)=(sinh^3(X))/(X^5+2 X^3)
because
f(0)=0/0
f'(0)=0/0
etc
thanks
First You can verify the 'identity'...
$\displaystyle \displaystyle f(x)= \frac{\sinh^{3} x}{x^{5}+2 x^{3}} = \frac{1}{2}\ \frac{1}{1+\frac{x^{2}}{2}}\ \frac{\sinh^{3} x}{x^{3}}$ (1)
... and the observe that is...
$\displaystyle \displaystyle \frac{1}{2}\ \frac{1}{1+\frac{x^{2}}{2}} = \frac{1}{2}\ \sum_{n=0}^{\infty} (-1)^{n}\ (\frac{x^{2}}{2})^{n}$ (2)
$\displaystyle \displaystyle \frac{\sinh x}{x} = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n+1)!}$ (3)
... so that You have all the elements to compute all the derivatioves of f(*) in 0...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$