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

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

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 $C \in R^+$?