# Sum power series not continuous unit circle

Let us consider the sequence $$(a_n)_{\geq 1} = \left(\frac{\cos(\sqrt{n})}{n^{\frac{3}{4}}} \right)_{n \geq 1}$$ and the associated power series $$\sum_{n \geq 1} a_n z^n$$. It is easily checked that the radius of convergence of this series is $$1$$. Furthermore, one can prove this power series converge also on the unit circle. Let me sketch an argument. It is in fact enough to prove that the power series associated to the sequence $$(b_n)_{\geq 1} = \left(\frac{e^{i\sqrt{n}}}{n^{\frac{3}{4}}} \right)_{n \geq 1}$$ converges on the unit circle. Let $$z_0 = e^{i\theta} \in \mathbb{U}$$.

$$\bullet$$ If $$z_0 = 1$$. One must prove that the series $$\sum_{n \geq 1} \frac{e^{i\sqrt{n}}}{n^{\frac{3}{4}}}$$ is convergent. But an easy computation shows that: $$\frac{e^{i\sqrt{n}}}{n^{\frac{3}{4}}} = -2i.\left(\frac{e^{i\sqrt{n+1}} - e^{i\sqrt{n}} }{n^{\frac{1}{4}}} \right) + c_n,$$ where $$c_n$$ is the general term of an absolutely convergent series. The Abel's test then implies that the series $$\sum_{n \geq 1} \frac{e^{i\sqrt{n}}}{n^{\frac{3}{4}}}$$ is convergent.

$$\bullet$$ If $$z_0 \neq 1$$. One must prove that the series $$\sum_{n \geq 1} \frac{e^{i\sqrt{n}+ i n\theta}}{n^{\frac{3}{4}}}$$ is convergent. The same computation as before shows: $$\frac{e^{i\sqrt{n} +in \theta}}{n^{\frac{3}{4}}} = \frac{1}{e^{i\theta}-1}.\left(\frac{e^{i\sqrt{n+1} + i(n+1)\theta} - e^{i\sqrt{n} +in \theta} }{n^{\frac{3}{4}}} \right) + d_n,$$ where $$d_n$$ is the general term of an absolutely convergent series. The Abel's test again implies that the series $$\sum_{n \geq 1} \frac{e^{i\sqrt{n} + in \theta}}{n^{\frac{3}{4}}}$$ is convergent.

Let me denote by $$F$$ the sum of the power series on the unit disk. The accepted answer to this question and the comments below seem to imply that $$F$$ is not continuous on the unit circle. However the arguments developped in the answer seem to me rather involved and roundabout (asymptotic estimations of the Laguerre polynomials, computations with Mapple, etc). I would like to now if there is a relatively simple and direct argument which shows that $$F$$ is not continuous on the unit circle.