$$D=\{z\in\Bbb C:|z|<1\}$$. Let $$f(z)=\sum\limits_{n=0}^\infty a_n z^n(a_n,z\in\Bbb C)$$be a power series, and the radius of convergence of $$f(z)$$ is $$1$$, $$\sum\limits_{n=0}^\infty a_n =s$$. we cannot conclude that

$\lim_{D\ni z\to1 }f(z)= s.$

In 1916, Sierpiński constructed 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 $$z=1$$.

Sierpiński 的例子很复杂, 在一本法文书上可以找到.

For odd $$n$$ let  $$p_n = 1\cdot 3\cdot 5\cdots n$$, For even $$n$$ set $$p_n=2p_{n-1}$$. Define

$f(z)=\sum_{n=1}^{\infty}\frac{(-1)^n}nz^{p_n}.$

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