I would like to expand a function as $$\frac{1}{x+1} = \frac{1}{x-1+2} = \frac{1}{x-1} \frac{1}{ 1+\frac{2}{x-1}} = \frac{1}{x-1} \left[ 1- \frac{2}{x-1} + \left(\frac{2}{x-1}\right)^2 + \cdots \right]$$
I tried
Series[1/(x + 1), {x, Infinity + 1, 3}]
apprently it does not work. Is there any robust way to realize this kind of expansion? Assume $|x|$ is sufficiently large.