How do you find the radius of convergence of a Taylor series for a function centered at point $z_0$ without actually finding the Taylor series?
I know that we can use comparison test, ratio test or root test to find the radius when we find the Taylor series but how do you do this without finding the series?
For example, finding the radius of convergence for the Taylor series of $f(z) = z^i$ centered at $z_0=2$.
Thank you
Best Answer
If all else fails try the distance to the nearest singularity. For $z^i=\exp(i\ln z)$ we have a singularity at $z=0$, hence expect convergence radius $2$.