Let $$ f(x) = \frac{x-1}{x^2-3x+2}$$ Factorise the denominator and use partial fractions to write $f$ as the sum of two functions whose power series you know. Write down the power series centered at $a=0$ and determine the radius of convergence.
It specifically asks for partial fractions. I tried partial fractions using $A, B$, etc. and I got what I would have got had I just factored and cancelled out the same terms. What is this exactly asking? How do I do this?
It's asking you to create a power series representation of the rational function. By using partial fractions, you can split into two:
$$A(x-2) + B(x-1) = x-1$$ $$x = 2 \implies B = 1$$ $$x = 1 \implies A = 0$$
You get:
$${A \over x-1} + {B\over x-2} = {x-1\over x^2-3x+2} \implies {1\over x-2}= {x-1\over x^2-3x+2}$$
And you're right; you could have factored, but the question specifically asks for partial fractions for who knows what reason. Now, recall the power series:
$${1\over 1-r} = 1+r+r^2+\cdots \quad |r|<1$$
Then apply:
$${1\over x-2} = {1\over -2\left(1-\frac x2\right)}=-\frac 12 {1\over 1-\frac x2}$$
Rewrite as the geometric power series where $r = \frac x2$. Also remember the restriction of $|r|<1$, which helps you determine the radius of convergence.