Finding Power Series Representation for f(x) and Interval of Convergence

[chupe]

Homework Statement

Find the power series representation for the function f(x)=x/(x^2-3x+2) and determine the interval of convergence.

Homework Equations

The Attempt at a Solution

First I separate into partial fractions 2/(x-2) - 1/(x-1)

2/(x-2) = sum n=0 to infinity (x/2)^n
1/(x-1) = sum n=0 to infinity (x)^n

Now I just don't know how to make them one power series representation.

 

[dynamicsolo]

Temporarily, let x = 2u and add the two power series together. You will need to find the pattern of the coefficients for the general term. You can always switch back to using x as a variable at the end. (There may be something more clever to use, but I'm juggling something else just now...)

Watch out, by the way: 1 + x + x² + ... is 1/(1 - x) .

EDIT: thought about this a bit more -- the cleaner way to do this is to write the (\frac{x}{2})^{n} terms as \frac{x^{n}}{2^{n}} . You will still have a little work on sorting out the general term for the single series representation.