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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 \rightar …