$f(x) = \frac{4 + x}{2 + x - x^2}$, 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?