Talk:Convergence of Taylor Series of Function Analytic on Disk

I think that all refactoring has been done now, so the refactor template could be removed. Can I do that?. --Ivar Sand (talk) 09:58, 14 November 2019 (EST)

Background

In real analysis we have the root test and the comparison test. But they only say how to determine the radius of convergence. They do not explain the actual value of the radius of convergence or say anything about how far a series reaches. For instance, the Taylor series for the function 1/x developed at x=1 has radius of convergence equal to 1 and therefore reaches the singularity at x=0. But is it true in general that the Taylor series for a function reaches the function's closest singularity? As a student, I asked myself this question, but found that this was not talked about in real analysis. The reason for this is the fact that the desired theorem does not belong in real analysis but in complex analysis. I still felt that such a result was lacking in real analysis and started proving it using a real analaysis perspective. The result is the theorem on the proof page.

It needs to be pointed out that this proof has a student's perspective. It is made for students who know real analysis but have little knowledge of complex analysis. It may be of no interest to the rest of the mathematical community. There is another example of a proof with a student's perspective here. --Ivar Sand (talk) 06:50, 2 December 2019 (EST)