Does there exist a power series of radius of convergence $R$ that uniformly converges on the open disc of radius $R$?

Intuitively, I do not think this is the case since there would be a singularity at some point on $|z|=R$, and so when we get near there, the series tends towards infinity and there is no way for the series to converge uniformly. But I have heard from others that the answer is that such a power series does in fact exist.

  • 1
    $\begingroup$ Sure $\sum z^n/n^2, n \ge 1$; the singularity at $1$ is not isolated; function is continuous on the unit disc; with more care we can make it infinitely differentiable on the circle too (as one variable there or as limit from inside of derivatives) $\endgroup$
    – Conrad
    Oct 24, 2020 at 18:31

1 Answer 1


The power series $\displaystyle\sum_{n=1}^\infty\frac{z^n}{n^2}$ has radius of convergence $1$ and it converges uniformly on $\overline{D(0,1)}$; in particular, it converges uniformly on $D(0,1)$.

  • $\begingroup$ Is the best way to see this from the $M$-test? $\endgroup$
    – Vasting
    Oct 24, 2020 at 18:52
  • $\begingroup$ That's how I would do it. $\endgroup$ Oct 24, 2020 at 18:52

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