Convergence of complex log series on the boundary

[polydigm]

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

 

[Citan Uzuki]

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.