## $f(x) = \frac{4 + x}{2 + x – x^2}$, calculate $f^{(9)}(1)$

$$f(x) = \frac{4 + x}{2 + x – x^2}$$, calculate $$f^{(9)}(1)$$, where $$f^{(9)}$$ is the $$9$$-th derivative of $$f$$.

Domain of $$f$$ is $$\mathbb{R} – \{-1, 2\}$$. I’ve got that $$f(x) = \frac{1}{1 – (-x)} + \frac{1}{1 – \frac{1}{2}x} = \sum_{n=0}^\infty ((-1)^n + 2^{-n})x^n$$, but there is a problem that $$\frac{1}{1 – (-x)} = \sum_{n=0}^\infty (-1)^nx^n$$ is convergent only for $$|x| < 1$$, so not for $$1$$. How can I go about this?