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 $$1$$, and $$f$$ is convergent at every point of the unit circle. If $$f(z)=0$$ for every $$|z|=1$$, then

$a_n=0$

for all nonnegative integer $$n$$.

It seems that this is a particular case of an old Theorem from Cantor (1870), called

Cantor’s uniqueness theorem. If, for every real $$x$$,

$\lim_{N \rightarrow \infty} \sum_{n=-N}^N c_n e^{inx}=0,$

then all the complex numbers $$c_n(n\in\Bbb Z)$$ are zero.

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