Radius of convergence of power series of log z about a point

Question

I would like to determine the radius of convergence of the power series of log z expanded at -4+3i. I'm not sure how to tackle this - I've tried calculating the power series but I don't see what test or theorem I should use after that to find the radius of convergence. Can anyone point me in the right direction? Thanks!

Answer 1

$\frac {d}{dz} \ln z = \frac 1z = \frac 1{a+ (z-a)} = \frac 1a \sum (\frac {(a-z)}{a})^n\\ \ln z = \int \frac 1a \sum (\frac {(a-z)}{a})^n dz = \sum \frac 1n(\frac {(a-z)}{a})^n+\ln a$

radius of convergence:

Ratio test or root test will both give you the same thing.

$|\frac {a-z}{a}| < 1\\ |a-z| < |a|$

Answer 2

There is a branch of $\log z$ holomorphic in $D(a,|a|)$, but not in $D(a,r)$ for any $r>|a|$. So basic complex analysis says the radius of convergence is $|a|$.