I want to find the asymptotic series of $g(x)=\frac{1}{x-1}, x>1$ in negative powers of $x$ while $x \to +\infty$, i.e. the sequence of coefficients $c_0, c_1, c_2, \dots$ such that $g(x) \sim c_0+\frac{c_1}{x}+\frac{c_2}{x^2}+\dots, x \to +\infty$
I have done the following:
We have that $\frac{1}{1-y}=1+y+y^2+\dots$, while $y \to 0^+$
So $g(x) \sim (-1) \sum_{n=0}^{\infty} x^n$.
But how can we find an asymptic series in negative powers?
You can write $$\frac 1{x-1}=\dfrac{\frac 1x}{1-\frac 1x}\\ =\frac 1x\left(1+\frac 1x+\frac 1{x^2}+\frac 1{x^3}\ldots\right)\\ =\frac 1x+\frac 1{x^2}+\frac 1{x^3}+\frac 1{x^4}\ldots$$ This converges for $|x| \gt 1$