Does there exist a power series such that points on its circle of convergence alternate infinitely often between convergence and divergence?

Asked Jul 11 '22 at 15:29

Active Jul 11 '22 at 15:29

Viewed 27 times

Does there exist a power series $$S=\sum_{n=0}^\infty{a_nx^n}$$ with a radius of convergence of one such that points on its boundary circle in the complex plane alternate infinitely often between convergence and divergence?

I.e. for any $\theta_1, \theta_2 \in \mathbb{R}$ such that $S$ converges at $e^{\theta_1i}$ and $e^{\theta_2i}$, there exists $\theta_1<\alpha\in \mathbb{R}<\theta_2$ such that $S$ diverges at $e^{\alpha i}$, and vice versa.

asked Jul 11 '22 at 15:29

Joshiepillow

I like this question. May be rephrased to ask about convergence of Fourier series $\sum_{n=0}^\infty a_n e^{i n \theta}$ . – GEdgar Jul 11 '22 at 16:09

Does there exist a power series such that points on its circle of convergence alternate infinitely often between convergence and divergence?

0 Answers0