Radius of Convergence for analytic functions I know that the radius of convergence of any power series can be found by simply using the root test, ratio test etc. 
I am confused as to how to find the radius of convergence for an analytic $f$ such as 
$f(z)=\frac{4}{(z-1)(z+3)}$. 
I can't imagine that I would have to find the power series representation of this, find the closed form, and then use one of the convergence tests. I am fairly certain that the radius of convergence would have to do with the singularities at $1$ and $-3$, however, I can't find a formula for the radius of convergence..

The theory tells us that the radius of the expansion about $z=0$ is $1$, but you can see this computationally as well. It turns out, just by direct long division of power series, that:
$$
f(z)=-\sum_{i=1}^\infty\frac{3^i+(-1)^{i+1}}{3^i}z^{i-1}\,.
$$
