Interpretation of the radius of convergence

Question

What interpretation should one give to the radius of convergence of a series $\sum a_nz^n$ ?

I do know how it is mathematically defined and what it implies for convergence/divergence, but I'm having a hard time understanding what its meaning is behind the formulas.

Is there, for instance, a graphical interpretation, or something of that kind ?

Answer 1

First the radius of convergence is defined for a power series but not for whatever series. Then the power series converge inside the disk of convergence, the one centered in zero and having for radius the radius of convergence. And it diverges outside this disk.