I really do not understand the question.

The radius of convergence R is a number such that |x|<R the series converges absolutely. If |x|>R the series diverges. Now we say R = 0 if no such real number exists such that |x|<R implies convergence. And we say R = +oo if no such real number exists such that |x|>R implies divergence.

So are you asking to prove that ever power series has: positive radius of convergence, zero radius of convergence, or infinite radius of convergence?