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Emeritus
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 $\mathbb C$, a series is convergent if and only if it's absolutely convergent. So if you're talking about functions from $\mathbb C$ into $\mathbb C$, 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 $$\sum_{n=0}^\infty a_n \left( x-x_0 \right)^n$$ that's convergent and =f(x).