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$D=\{z\in\Bbb C:|z|<1\}$. Let $f(z)=\sum\limits_{n=0}^\infty a_n z^n(a_n,z\in\Bbb C)$ be a power series, the radius of convergence of $f$ is $1$, $\sum\limits_{n=0}^\infty a_n =s$.

  1. Give a power series $f$ such that $$\lim_{D\ni z\to1 }f(z)\ne s$$

  2. If $f$ is convergent at every point of the unit circle, is there such an power series $f$ ?

P.s. the problem is related to Abel's theorem

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In 1916, Sierpiński constructed an example of a power series with radius of convergence equal to $1$, also converging on every point of the unit circle, but with the property that $f$ is unbounded near $z=1$. The construction is not easy, and there may very well be more modern and perhaps more accessible examples. A reprint of the 1916 paper can be found here, p 282 (in French).

Sierpiński's example settles both your questions.


Added: This discussion on MO is highly relevant, and contains a simpler construction than Sierpiński's.

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