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The series I'm given is

$$\sum_{n=0}^\infty \frac{z^n}{1+z^n}$$

and I have to show it's holomorphic on the open unit disk. Now my thinking is to show that it is uniformly convergent on unit disk using Weierstrass M test. But not really sure where to go from there

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  • $\begingroup$ Man, I really don't know how to edit, and I end up spending more time on how to edit this that asking the question in the first place $\endgroup$ – Gragbow May 13 '17 at 15:02
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It's not uniformly convergent on the unit disc: each term has a pole on the unit circle. But you should be able to prove that the series in uniformly convergent on the disc with centre $0$ and radius $r$, for any $r<1$. That will suffice to prove the sum holomorphic.

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  • $\begingroup$ Sorry, it's on the open unit disk. My apologies $\endgroup$ – Gragbow May 13 '17 at 15:19
  • $\begingroup$ @DanielMcElroy Daniel, it is the open disk that we are discussing. But the series doesn't uniformly convergence on the open disk $|z|<1$. It does converge, however, on each disk $|z|\le \rho<1$. $\endgroup$ – Mark Viola May 15 '17 at 6:10

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