# Region of convergence: Complex vs Real valued power series.

The real valued power series:

$$\sum_{n=0}^\infty a_n(x-x_0)^n$$

converges when: $$|x-x_0| and diverges when $$|x-x_0|>R$$. I want to find the region of convergence for the complex valued series:

$$\sum_{n=0}^\infty a_n(z-z_0)^n$$

Which, is this not obviously $$|z-z_0|? What method to use to prove the result?

• By basic properties of power series the radius of convergence is $R$ and the circle of convergnec is precisely $\{z: |z-z_0| <R\}$. – Kavi Rama Murthy Oct 16 '18 at 23:28
• @KaviRamaMurthy Correct, but I believe the idea is to give it in terms of $R$ of the real valued series. Although I may be misreading the question. – Dole Oct 17 '18 at 3:04