Gamma functions approaches solely n to a negative power

Question

I have read $[x^n] (1-x)^\alpha\sim \frac{n^{-\alpha-1}}{\Gamma(-\alpha)}$ as $n\rightarrow\infty$ where $[x^n]$ means coefficient of $x^n$ in what follows. But I have never seen a proof of this assuming can take the symbol $\sim$ to also mean approximately or approaches but I have yet to see any proof. By letting $\alpha \rightarrow -\alpha$ in a series formula I have verified, I use $$(1-x)^\alpha= \frac{1}{\Gamma(-\alpha)} \sum_{n=0}^\infty \frac{\Gamma(n-\alpha)}{n!}x^n,|x|<1$$ So this and the above implies $\frac{\Gamma(n-\alpha)}{n!}\rightarrow n^{-\alpha-1} ,n\rightarrow\infty$ And I certainly cannot reconcile this. So what is the discrepancy ? Is it an error in the text where I read this ?