(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If f(z) = [itex]\sum[/itex] a_{n}(z-z_{0})^{n}has radius of convergence R > 0 and if f(z) = 0 for all z, |z - z_{0}| < r ≤ R, show that a_{0}= a_{1}= ... = 0.

2. Relevant equations

3. The attempt at a solution

I know it is a power series and because R is positive I know it converges. And if f(z) = 0 then the sum itself would be 0 which would mean that each term must add to 0. This doesn't necessarliy imply that each coefficient a_{i}is 0 though because they could alternate a_{1}= 1, a_{2}= -1, a_{3}= 2, a_{4}= -2 etc... Am I right in thinking this? And if so, I'm not sure how to go from there... Unless f(z) = 0 for all z implies that |z - z_{0}| ≠ 0 somehow then for f(z) to equal 0, the a_{i}would have to equal 0? Any help would be appreciated, thanks!

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# Complex Power Series Radius of Convergence Proof

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