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Conditions:

  1. $\lim\limits_{n\to\infty}a_n=0$,
  2. $\forall z\in\partial\mathbb{D}$, $\sum_{n=1}^{\infty}a_nz^n$ diverges. Here $\mathbb{D}=\{w\in\mathbb{C}:|w|<1\}$.

I tried to write $a_n=x_n+iy_n$, where $(x_n)_{n\in\mathbb{N}},(y_n)_{n\in\mathbb{N}}$ are real sequences and use the second condition ( such as taking $z=\pm1$ ), but I found it hard to get more information about $(a_n)_{n\in\mathbb{N}}$. Trying more value of $z$ seems hard to see something meaningful and I felt bewildered.
Are there any results about this problem? Thanks for explanation in advance.
This question comes from my classmate, so I'm not sure whether there is some clear answer.

Confusion
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    Would the closer deign to clarify how the linked posts answer the question? Particularly, where does anyone mention the condition $\lim_{n \to \infty} a_n = 0$? – Adayah Sep 23 '22 at 13:55
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    @Adayah Lusin's example works because all coefficients are less than $m^{-1/2}$ (the spacing of $g_m$ and $h_m$ is exactly so that every power appears at most once). By the way, CommunityBot means that the OP closed the question himself. – Klaus Sep 23 '22 at 15:18
  • @Klaus Thanks for the clarification of the example and the way to interpret the presence of Community Bot on the list of close voters. The second is quite weird, if I may add. – Adayah Sep 23 '22 at 16:25
  • Thanks for comments! – Confusion Sep 24 '22 at 05:28

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