$D=\{z\in\Bbb C:|z|<1\}$. 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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.