- #1

polydigm

Gold Member

- 5

- 0

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?

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: