# Can a power series uniformly converge on open disc?

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.

• 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) Oct 24, 2020 at 18:31

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 $$D(0,1)$$.
• Is the best way to see this from the $M$-test? Oct 24, 2020 at 18:52