Factor out a negative power before expanding

For $x\ne0$, $(2+1/x)^{-2}=x^2(1+2x)^{-2}$.

$(1+2x)^{-2}=1+(-2)(2x)+\dfrac{(-2)(-3)}{2!}(2x)^2+\dfrac{(-2)(-3)(-4)}{3!}(2x)^3+O(x^4)=1-4x+12x^2-32x^3+O(x^4)$. Thus $(2+1/x)^{-2}=x^2-4x^3+12x^4-32x^5+O(x^6)$.

Require $|2x|<1$ and $x\ne0$, so $-1/2<x<1/2$ with $x\ne0$. The rewritten function has a removable extension at zero, but the original expression is undefined there.