Power Series and Absolute summability

Question

I have a quick question regarding power series. Let $\psi_1, \psi_2, \ldots$ denote the real-valued coefficients of a power series. I would like to see a proof (or a counterexample) to the following result:

$$\left| \sum_{j=0}^{\infty} \psi_j z^j \right |< \infty \quad \forall \: z \in \mathbb{C} \textrm{ such that } |z| \leq 1 \implies \sum_{j=0}^{\infty} |\psi_j| < \infty.$$

If the result is true: could one replace the complex plane $\mathbb{C}$ by the real line $\mathbb{R}$?

Thanks very much for your help.

Answer 1

Take $\psi_n=1$ for all $n\in\mathbb N$. Then $\sum_{n=0}^{\infty} z^n$ converges for all $z\in\mathbb C$ such that $\left|z\right|<1$.

But the power series diverges at all point of the unit circle (for $z^n$ doesn't have limit $0$).

You can compute other examples, where the power series converges only at certain points, or everywhere but a few points...

Answer 2

There is a power series converging uniformly but not absolutely on the unit circle (Hardy's example; see the book by Landau-Gaier MR 88d:01046, p. 68, or Theorem 2.28 in a book by A. Sasane Algebras of holomorphic functions and control theory. Dover Publications (ISBN 978-0-486-47465-6). ix, 140 p. (2009). If you want real coefficients,just take the real parts of the coefficients.