I've been reading a complex analysis book which had an example showing [tex]\sum^\infty_{n=1}1/n \cdot z^n[/tex] is convergent in the open unit ball.(adsbygoogle = window.adsbygoogle || []).push({});

I'm now looking at the case when [tex]|z| = 1[/tex]. Clearly [tex]z = 1[/tex] is the divergent harmonic series, but i know this series is in fact convergent for all other [tex]|z| = 1[/tex].

In order to prove this, i need to be able to show that the related series [tex]\sum^\infty_{n=1}z^n[/tex] is bounded, whenever [tex]|z| = 1[/tex] and [tex]z \neq 1 [/tex],.

I can solve this problem when ever the argument of z is a rational multiple of pi, but other than that I'm stuck. Any help proving that this related series is bounded would be very helpful.

Thanks!

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# Showing a (complex) series is (conditionally) convergent.

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