$f(x) = \frac{4 + x}{2 + x – x^2}$, 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?