I'm trying to find the Radius of Convergence of a complex power series, and I can see that I can use the Ratio test (i.e. the limit of a_{n}/a_{n+1} exists). However, the first coefficient, a_{0} = 0 (but a_{n} is non zero for all other n).

Is it true that the radius of convergence of a power series from 0 to infinity is the same as the the radius of convergence for the same power series but changing to a sum from m to infinity. (for some arbitrary natural number m). That is, the radius of convergence is independent of where you chose to start your summation from?

This seems logical to me but I'm not 100% certain.

Thanks for any help!