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While starting complex analysis, we encountered complex power series and radii of convergence. My question is this. In real values cases, the power series has a radius of convergence, which is an open interval, but then we check the endpoints to see if it converges there as well to get an interval of convergence.

When we check a complex power series to see where it is analytic, what is the equivalent of checking the endpoints? Is it like a disk? And how would you do that?

A specific example is $\displaystyle\sum_{k=0}^{\infty}2^k(z-3)^{k!}$, I get radius is 1 centered at 3 using the ratio test, but how do we test endpoints?

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    $\begingroup$ The boundary of the domain of convergence is a disc, but frankly one rarely checks the boundary in the setting of (a first course in) complex analysis: having convergence at an endpoint does not imply analyticity there. $\endgroup$ – KCd May 5 '18 at 2:03
  • $\begingroup$ The boundary of the disk is just a circle. $\endgroup$ – Henning Makholm May 5 '18 at 2:14

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