The thing is that the convergence radius is exclusively limited by singularities. If you have any analytic function and develop it at a certain point into a powerseries, then the distance to the nearest singularity is the radius of convergence. For example if you develop log at the point 0.3 then its radius of convergence is 0.3 because there is a singularity at 0.

On the real axis however you dont see all the singularities lurking in the complex. For example you can have an analytic function that has no singularities on the real axis, however it is not entire because regardless at which point you develop there is the singularity in the complex that limits the radius of convergence.

On the real axis however you dont see all the singularities lurking in the complex. For example you can have an analytic function that has no singularities on the real axis, however it is not entire because regardless at which point you develop there is the singularity in the complex that limits the radius of convergence.