How do I find the series' formula knowing its generating function: $f(x) = \frac{x}{6x^2-5x+1}$

Question

$f(x) = \frac{x}{6x^2-5x+1}$ is the generating function of a series. How do I go about finding the formula of this series?

I started by finding the partial fractions: $$f(x) = \frac{1}{2x-1} + \frac{-1}{3x-1}$$ How do I proceed from here?

Answer 1

hint

As recomended, by Leon,

If $ |X|<1 $, then

$$\frac {1}{1-X}=1+X+X^2+...=\sum_{k=0}^{+\infty}X^k$$

So, if $ |3x|<1 $,

$$f(x)=\frac{1}{1-3x}-\frac{1}{1-2x}=$$

$$\sum_{k=0}^{+\infty}((3x)^k-(2x)^k)$$