Prove that f(x) = Σ (k=0 to ∞) a_k x^k has a positive radius of convergence.
Can someone give me some help on where to start with this?
The radius of convergence R is a number such that |x|<R the series converges absolutely. If |x|>R the series diverges. Now we say R = 0 if no such real number exists such that |x|<R implies convergence. And we say R = +oo if no such real number exists such that |x|>R implies divergence.
So are you asking to prove that ever power series has: positive radius of convergence, zero radius of convergence, or infinite radius of convergence?
Thanks. Unfortunately, the question was just as confusing to me. My thought was the same as yours that R had conditions and was not always positive. I thought maybe there was something that I was not understanding. I'm assuming now that I'm supposed to solve it for R>0. Who knows with this text!