Hi, I'm new here. I am curious that why a power series must have a radius of convergence? I mean, even in a complex plane, there is always a so-called convergent radius for a power series. Is it possible that a power series is convergent for a certain range in one direction, and for an apparent shorter/longer range in some other direction? So far all the text books I read do not give lessons over this question. Any answer or hint or instruction will be much appreciated.
The answer is no, it can't have a larger range of convergence in a different direction. The relevant theorem is the Cauchy-Hadamard theorem. Lots of links on the internet, one of which is: http://eom.springer.de/c/c020870.htm
It is, basically, an application of the "ratio test". If [itex]f(z)= \sum a_n(z- z_0)^n[/itex]] then the series converges, absolutely, as long as [tex]\lim_{n\to\infty}\frac{|a_{n+1}(z- z_0)^{n+1}|}{|a_n (z- z_0)^n|}[/tex][tex]= |z- z_0|\lim_{n\to\infty}\frac{a_{n+1}{a_n}|< 1[/tex] and diverges if that limit is larger than 1. As long as [tex]\lim_{n\to\infty}\frac{a_{n+1}}{a_n}= A[/tex] exists, then we have that the power series converges for [tex]|z- z_0|< \frac{1}{A}[/tex] and diverges for [tex]|z- z_0|> \frac{1}{A}[/tex] You can get the same result by using the root test instead of the ratio test: [itex]\sum a_n (z- z_0)^n[/itex] converges absolutely as long as [tex]\lim_{n\to\infty}\left(a_n(z- z_0)^n)^{1/n}= \left(\lim_{n\to\infty}\sqrt[n]{a_n}\right)|z- z_0|[/tex] is less than 1.