Taylor Expansion where the derivatives are undefined?

[pandaBee]

Homework Statement

Expand x/(x-1) at a=1
The book already gives the expansion but it doesn't explain the process. The expansion it gives is:
x/(x-1) = (1+x-1)/(x-1) = (x-1)^(-1) + 1

Homework Equations

The Attempt at a Solution

I've already solved for the Mclaurin expansion for the same polynomial which gives me -x - x^2 - x^3 - x^4 - ...
The derivatives all have a power of (x-1) as its denominator therefore all of its derivatives are undefined. I don't know any other way to expand polynomials other than term by term, I've been staring at

x/(x-1) = (1+x-1)/(x-1) = (x-1)^(-1) + 1

But it doesn't make sense. All they did was just split the fraction up into two parts.

 

[micromass]

Are you sure you don't want the Laurent expansion instead of the Taylor expansion?

 

[pandaBee]

Are you sure you don't want the Laurent expansion instead of the Taylor expansion?

Sorry about that, yes I believe that is the case.

 

[Office_Shredder]

Are you confused about how (x-1)^-1 + 1 is a Laurent series?

 

[pandaBee]

Are you confused about how (x-1)^-1 + 1 is a Laurent series?

I'm confused about the method used to derive it. As in I don't understand the method of the reasoning behind it.

Also I don't see how they can come up with that answer when the function x/(x-1) is undefined around x=1.