Quote by JamesGoh
What I undestand is that for all x values, xxo must be less than R (radius of convergence) in order for f to be analytic at xo.

What you're saying here would imply that the truth value ("true" or "false") of the statement "f is analytic at x
_{0}" depends on the value of some variable x. It certainly doesn't. It depends only on f and x
_{0}. (What you said is actually that if xx
_{0}≥R, then f is not analytic at x
_{0}).
I'm a bit surprised that your definition says "converges absolutely". I don't think the word "absolutely" is supposed to be there. But then, in [itex]\mathbb C[/itex], a series is convergent if and only if it's absolutely convergent. So if you're talking about functions from [itex]\mathbb C[/itex] into [itex]\mathbb C[/itex], then it makes no difference if the word "absolutely" is included or not.
What the definition is saying is that there needs to exist a real number R>0 such that for all x with xx
_{0}<R, there's a series [tex]\sum_{n=0}^\infty a_n \left( xx_0 \right)^n[/tex] that's convergent and =f(x).
I like Wikipedia's definitions by the way.
Link.