When we apply binomial expansion $(1+z)^\alpha = \sum \dbinom{\alpha}{n} z^n$ for $z \in \mathbb{C}$ with $|z|<1$, we have:
If $\alpha=-1$, then $(1+z)^{-1} = \sum \dbinom{-1}{n} z^n$. On the other hand $(1+z)^{-1} = (1-(-z))^{-1} = \sum (-1)^n z^n$.
But Taylor expansions about $0$ are unique - and clearly these do match? What went wrong?
Nothing went wrong because we have
$$ { -1 \choose n } = \frac{(-1)(-2)\cdots(-n)}{n!} = \frac{(-1)^n n!}{n!}= (-1)^n.$$