Laurent Series of a Rational Polynomial

Question

I am trying to compute the Laurent expansion of $\frac{x}{x-1}$ around $\alpha=2$. I wrote it out as a partial fraction expansion so we have: $$ \frac{x}{x-1} = 1 + \frac{1}{x-1} = 1-\frac{1}{1-x} $$ And so we should have the Laurent expansion of this by the geometric series obtaining: $$ 1-\sum_{n=0}^\infty (x-2)^n = -(x-2)-(x-2)^2-\cdots $$ But this is wrong, according to the textbook I am working with the correct answer should be: $$ 2-(x-2)+(x-2)^2-(x-2)^3+\cdots $$ I don't know where exactly I've gone wrong either, I suspect I've made a mistake in computing the expansion at $\alpha=2$ because I know that when $\alpha=0$ this would have given the correct result. I'd just like to understand where I went wrong, I think I am misunderstanding something critical.

Answer 1

$1+\dfrac1{x-1} =1+\dfrac1{1+(x-2)}=1+1-(x-2)+(x-2)^2-(x-2)^3+...$

Answer 2

You went wrong when you wrote that$$\frac1{1-x}=\sum_{n=0}^\infty(x-2)^n.$$The sum of this series is actually$$\frac1{1-(x-2)}=\frac1{3-x}.$$