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The real valued power series:

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

converges when: $|x-x_0|<R$ 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 $|z-z_0|<R$? What method to use to prove the result?

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  • $\begingroup$ By basic properties of power series the radius of convergence is $R$ and the circle of convergnec is precisely $\{z: |z-z_0| <R\}$. $\endgroup$ – Kavi Rama Murthy Oct 16 '18 at 23:28
  • $\begingroup$ @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. $\endgroup$ – Dole Oct 17 '18 at 3:04

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