Power series divergence (real analysis)

Question

Show that if a power series diverges at $x_0$ then it must also diverge when $\lvert x\rvert > \lvert x_0\rvert$ or provide a counterexample.

I feel like there is a counterexample for some kind of alternating series, but I'm not sure.

Answer 1

This follows directly from the theorem of radius of convergence. If you can prove that there exists an R (radius of convergence) such that for any $\lvert x \rvert>\lvert R \rvert$ the series $\displaystyle \sum_{n=0}^\infty a_nx^n$ diverges then you have the proof you are looking for. I won't go into the details here since the proof for radius of convergence has already been answered: Proof of the "Radius of Convergence Theorem"