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I came up with a question for a proof of Abel's limit theorem (P.41 Complex Analysis by Ahalfors). Help from those who have this book is appreciated.

The theorem states that if a power series $a_0+a_1 z+ a_2 z^2+...$ has the convergence radius $R=1$ and converges at $z=1$, then a function $f(z)=a_0+a_1 z+ a_2 z^2+... $approaches to $f(1)$ as $z\to 1$ in a such way that $|1-z|/(1-|z|)$ is bounded.

In the proof the author defines a partial sum $s_n=a_0+a_1+a_2+...$ and say "$s_n z^n \to 0$". My question is that: 1) "$s_n z^n \to 0 $" as $n \to 0$ or $z \to 1$? 2) Why does it tend to $0$?

Thanks in advance.

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    @HaraldHanche-Olsen, Lukas I didn't take into consideration that |z|<1. Thank you very much!2012-11-01

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