1. ## Taylor's series

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

2. First You can verify the 'identity'...

$\displaystyle f(x)= \frac{\sinh^{3} x}{x^{5}+2 x^{3}} = \frac{1}{2}\ \frac{1}{1+\frac{x^{2}}{2}}\ \frac{\sinh^{3} x}{x^{3}}$ (1)

... and the observe that is...

$\displaystyle \frac{1}{2}\ \frac{1}{1+\frac{x^{2}}{2}} = \frac{1}{2}\ \sum_{n=0}^{\infty} (-1)^{n}\ (\frac{x^{2}}{2})^{n}$ (2)

$\displaystyle \frac{\sinh x}{x} = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n+1)!}$ (3)

... so that You have all the elements to compute all the derivatioves of f(*) in 0...

Kind regards

$\chi$ $\sigma$