I am reading Gamelin's Complex Analysis book, and stumbled upon a statement while working on a question. the question asks to show that $\sum \frac{z^n}{n}$ is not uniformly convergent for $|z| < 1$. However, the back of the book says if it converges uniformly for $|z| < 1$ then it also converges uniformly for $|z| \leq 1$. Is this a typo or am I missing something? I thought if it converges uniformly $|z| <1$, then it converges absolutely for $|z| \leq 1$, but not necessarily uniformly. am I wrong?

  • $\begingroup$ Absolute convergence implies uniform convergence here since $|z| \le 1$ so we can disregard it in estimates, so if $\Sigma{|a_n|} < \infty$ estimates become uniform $\endgroup$ – Conrad Apr 16 at 18:41
  • $\begingroup$ so, is the statement correct?, I mean, if it is uniformly convergent for $ |z| < 1$ then it must be uniformly convergent also for $|z| \leq 1$? $\endgroup$ – dirichlet237 Apr 16 at 18:43
  • $\begingroup$ I think so but it doesn't imply absolute convergence unless the coefficients are positive as there are easy counterexamples with oscillating coefficients; here though an easy proof works since if you take $r=1-\frac{1}{n}, n \le m \le 2n$, you can easily show the partial sum not that small which contradicts uniform convergence when $n$ big $\endgroup$ – Conrad Apr 16 at 18:55

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