If the power series $\sum a_nz^n$ converges at $3+4i$ then find radius of convergence of the series 
If the power series $\sum a_nz^n$ converges at $3+4i$ then find radius of convergence of the series.

I can conclude that radius of convergence is either $\ge 5$ or $\le 5$.But how to find it?
Please help
 A: You cannot find the radius of convergence exactly here.  The center of expansion is 0, let $\rho$ be the radius of convergence.  The relevant theorem says that the series converges (absolutely) for $|z| < \rho$ and diverges for $|z| > \rho$.  Therefore you know that the series is only allowed to converge for $|z| \leq \rho$ (but there is no guarantee of convergence on the circle of course).  Since it converges when at $z=3+4i$, then is that the we get that $5 = |3+i4| \leq \rho$.  And that's the best that you can say about $\rho$.  It could be that it is 5, it could also be that it is 6.1, and it could even be infinity.  For example the series for $e^z$ converges at $3+4i$, and in fact it converges everywhere.  So the question, as stated, does not have enough information to state precisely what $\rho$ is.
On the other hand if you knew that the series converges but not absolutely at $3+4i$, then you could conclude that $\rho=5$.
A: The series converges at $z$ if $|z|$, which is the distance from $0$ to $z$, is less than the radius of convergence, and diverges if if $|z$ is more than the radius of convergence. If $|z|$ is exactly equal to the radius of convergence, then it may converge or diverge depending on which series you've got and which point on the boundary of the circle is $z$.
If $z= 3 + 4i$ then $|z| = \sqrt{3^2+4^2} = 5.$ So the radius of convergence must be $\ge 5$. The radius might be exactly $5$ or some finite number more than $5$ or $\infty$, depending on which series it is.
