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Convergence of complex log series on the boundary

  1. Mar 24, 2012 #1

    polydigm

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    The radius of convergence of [itex]\sum\limits_{k=1}^\infty\displaystyle\frac{z^n}{n}[/itex] is 1. It converges on all of the boundary [itex]\partial B(0,1)[/itex] except at [itex]z=1[/itex]. One way of looking at this is to analyse [itex]\sum\limits_{k=1}^\infty\displaystyle\frac{\cos n\theta}{n}+\frac{\sin n\theta}{n}[/itex]. You can see the [itex]\theta = 0[/itex] solution is just the harmonic series which doesn't converge, but that the [itex]\theta = \pi[/itex] solution does converge.

    The latter sum I am yet to figure out how to handle in general. Does anyone know of a good online reference that covers this or a relevant book I may be able to find in my uni library?
     
    Last edited: Mar 24, 2012
  2. jcsd
  3. Mar 24, 2012 #2
    Not sure about a reference, but I do have a suggestion for an approach to the problem. Instead of restricting attention to the boundary circle, you can try to find a general formula for partial sums on the whole closed unit disk. This can be done by finding a formula for the derivative of these partial sums with respect to z (which is a geometric series), then integrating it. Then apply the dominated convergence theorem to the integral to obtain the value of the series on the whole closed unit disk.
     
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