The power series $f(z)=\sum_{n=0}^{\infty}a_nz^n$ has convergent radius $r>0$, so $f(z)$ is analytic in $|z|<r$. MOREOVER, we suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is continuous on the closed disk $|z|\leq r$. My question is that: Does the series $\sum_{n=0}^{\infty}a_nz^n$ converge uniformly on the closed disk $|z|\leq r$? In my opinion, I can give a example such as: the series $$f(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^2}$$ has radius $r=1$ and converges uniformly on the closed unit disk $|z|\leq1$. I do not know whether my question is right or wrong. Any hints will welcome!

  • $\begingroup$ That's an example where the series converges uniformly. But it does not answer the question! To answer the question you either have to give an example where $f$ is continuous on $|z|=r$ but the series does not converge uniformly, or show that continuity implies uniform convergence. $\endgroup$ Commented Jun 4, 2018 at 12:42
  • $\begingroup$ The example I gives implies my question maybe right! $\endgroup$
    – Riemann
    Commented Jun 4, 2018 at 12:46
  • $\begingroup$ If you say so. "maybe right" or not, it's not true: In fact assuming $f$ is continuous on $|z|=r$ does not imply uniform convergence. A counterexample is not quite trivial... $\endgroup$ Commented Jun 4, 2018 at 13:07
  • $\begingroup$ I do not know the count-example, can you give me the reference or the website! Thanks a lot! $\endgroup$
    – Riemann
    Commented Jun 4, 2018 at 13:10
  • $\begingroup$ Goggling "uniformly convergent fourier series" led to a wikipedia article that says this: "There exist continuous functions whose Fourier series converges pointwise but not uniformly; see Antoni Zygmund, Trigonometric Series, vol. 1, Chapter 8, Theorem 1.13, p. 300." $\endgroup$ Commented Jun 4, 2018 at 13:17


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