<s>[INST] How to express $f(x)=\frac{1}{x-1} + \frac{1}{x-3}$ as a power series? I am working through my Calculus 2 course problems, and I have reached..

Express the function $f(x)=\frac{2x-4}{x^2-4x+3}$ as the sum of a power series by first using partial fractions.

I can do the partial fractions..
$$\frac{2x-4}{x^2-4x+3}=\frac{2(x-2)}{(x-1)(x-3)} =\frac{A}{x-1}+\frac{B}{x-3}$$
$$2(x-2)=A(x-3)+B(x-1)$$
$$\to -2=-2A \to1=A$$
$$\to2=2B\to 1 = B$$
$$\frac{1}{x-1} + \frac{1}{x-3}$$
I am not sure how to approach creating a power-series with two fractions like this. 
 [/INST] As written in the comments for your question, for $\vert r\vert<1$ we know that:
$$ \sum\limits_{n=0}^\infty r^n= \frac{1}{1-r} $$
You can rewrite for all $a\neq 0$:
$$ \frac{1}{x-a}=-\frac{1}{a} \cdot \frac{1}{1-\frac{x}{a}}= -\frac{1}{a} \sum\limits_{n=0}^\infty \Big( \frac{x}{a} \Big)^n= -\sum\limits_{n=0}^\infty \frac{x^n}{a^{n+1}} $$
Which is true for all $\vert x\vert<\vert a\vert$. And you can apply that to your fractions.
</s>