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.
No comments:
Post a Comment