Fredrik

What you're saying here would imply that the truth value ("true" or "false") of the statement "f is analytic at x₀" depends on the value of some variable x. It certainly doesn't. It depends only on f and x₀. (What you said is actually that if |x-x₀|≥R, then f is not analytic at x₀).

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 |x-x₀|<R, there's a series [tex]\sum_{n=0}^\infty a_n \left( x-x_0 \right)^n[/tex] that's convergent and =f(x).

I like Wikipedia's definitions by the way. Link.