Friday, 3 March 2017

calculus - Do their exist power series with non circular regions of convergence?



So far just about any series of the form




i=0(aixi)



Has tended to have a circular disk of convergence (of some radius, sometimes even 0).



Is there a reason this is always the case? Do there exist power series with say a lemniscate of convergence or an oval of convergence, etc... Or is there a clear reason why a power series must converge over a disk of some radius (possibly 0)


Answer



The Cauchy-Hadamard Theorem tells us that for every power series over C, there is an R[0,] such that the series converges for |x|<R and diverges for |x|>R. So yes, it is always a circle of convergence.



Edit: Yes, it can go both ways on the boundary. Usually that's still called a circle of convergence, tough.


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