Radius of convergence of a complex power series? Let's say I have a series which converges in a radius $R$ about point $a$. Let's say I expand the same about point $b$. How does the radius of convergence change? 

My progress so far :- I obtain it to be $R-|a-b|$. However I clearly have points outside this radius where the series converges, such as $b$. What's wrong in this argument?
As Andre said in a comment, all you get is an inequality, because you have in general no idea how the function behaves at or outside the radius of convergence. For example, both $( z \mapsto \frac{1}{z-3} )$ and $( z \mapsto \frac{1}{z^2-9} )$ have a power series around $0$ with radius of convergence $3$, but the first has a power series around $-3$ with radius of convergence $6$ while the second does not even have a power series around $-3$. Also, both have a power series around $-1$ but different radius of convergence.

All you know is that if $b$ is within the original convergence disk around $a$, then we can guarantee that the power series around $b$ certainly will converge within a disk that fits in $a$, such as the disk of radius $R - |a-b|$.
Use |z-a|≧|z-b|-|a-b| to get the result as R+|a-b|