How to find the radius of convergence?

The function is $\dfrac {z-z^3}{\sin {\pi z}} $. How to find the radius of convergence in $ z=0 $?

It suffices to find which is the larger disk centered at the origin, i.e., $D(0,r)$, with $r$ maximum, in which this function is analytic. And as both numerator and denominator are entire functions, the fraction is analytic in those points where the denominator does not vanish. Unless numerator and denominator vanish simultaneously at some point, and the order of the zero of the denominator does not exceed the one of the numerator.

The radius of convergence is $2$ which is the nearest singularity to $0$. The point $z=2$ is a pole of order $1$.