# uniform convergence on interior

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?

• 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 – Conrad Apr 16 at 18:41
• so, is the statement correct?, I mean, if it is uniformly convergent for $|z| < 1$ then it must be uniformly convergent also for $|z| \leq 1$? – dirichlet237 Apr 16 at 18:43
• 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 – Conrad Apr 16 at 18:55