TITLE: counterexample to Abel's theorem
QUESTION [17 upvotes]: $D=\{z\in\Bbb C:|z|<1\}$. 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 $1$,   $\sum\limits_{n=0}^\infty a_n =s$.

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

REPLY [8 votes]: 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.