- #1
shoescreen
- 15
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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.
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!
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!