Let's consider the power series $\sum_{n = 0}^{\infty} a_nx^n $ with radius of convergence $1$. Moreover let's suppose that : $\sum_{n = 0}^{\infty} a_n= +\infty$. Then I would like to find a sequence $(a_n)_{n \in \mathbb{N}} \subseteq \mathbb{R}^{\mathbb{N}}$ that respect the above condition and such that :
$$\lim_{x \to 1, x < 1} \sum_{n = 0}^\infty a_nx^n \ne +\infty$$
First I've noticed that $a_n$ can't be a positive sequence, since if it was the case we would have for all $N$ :
$$\lim_{x \to 1} \sum_{n = 0}^\infty a_nx^n \geq \sum_{n = 0}^N a_n$$
Hence we need some of the $a_n$ to be negative. Moreover I need to use the assumption that the sum at $x = 1$ diverges, because if the sum at $x = 1$ converges then Abel's theorem says that the limit at $x \to 1$ and the sum of the power series at $x = 1$ are equal.
Thank you.