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use \mathbb{R} for real numbers

Does the following series converge? To which value?

Given $f(x)=A \cdot x^5+B \cdot x^8$ with:

  • $A \in \mathbb{R}^-$
  • $B \in \mathbb{R}^-$
  • $h(z) = w_0 + \sum_{n=1}^\infty h_n \cdot \frac{\left(z-f(w_0)\right)^n}{n!}$
  • $w_0=1$

and: $$h_n=\lim_{w \rightarrow w_0} \frac{d^{n-1}}{dw^{n-1}} \left[\left(\frac{w-w_0}{f(w) - f(w_0)}\right)^n\right]$$ i.e. $$h_n =\lim_{w \rightarrow 1} \frac{d^{n-1}}{dw^{n-1}} \left[\left(\frac{w-1}{A \cdot (w^5-1)+B \cdot (w^8-1)}\right)^n\right]$$

Does $h(z)$ converge? What can be said of $h(-C)^2$ with $C \in \mathbb{R}^+$?