Radius of convergence of function given centre of its power series?

Question

I understand how to calculate the Radius of Convergence of a Power Series or for example for a given function by seeing where it if analytic but how about in this case, what is the fastest way?

The function is : $$\frac{z^3}{1-z^4}$$ and its power series centre is: $$z=2-2i$$

I would personally get all the singularities then calculate the distances to the given centre and pick the smallest one as the radius of convergence.

Is there a faster way?

Thanks!

Answer 1

You only have to compute four distances. They are the distances from the four fourth roots of unity to $2-2i$.

Since they're evenly spaced around the unit circle, it's obvious that the closest is $e^{6\pi i/4}$. Because $2-2i$ is in the fourth quadrant.

The distance, and thus the radius of convergence, is $$\mid(2-2i)-(-i)\mid=\sqrt5.$$

Come to think of it, $1$ is equally close.