How to properly set up partial fractions for repeated denominator factors I was just trying to solve a problem that had the following item which I needed to split into separate generating functions:
$$\frac{x}{(1-2x)^2(1-5x)}$$
I had assumed I needed to split it into:
$$\frac{A}{1-2x} + \frac{B}{1-2x} + \frac{C}{1-5x}$$
But according to Wolfram Alpha it appears I had to split it into:
$$\frac{A}{1-2x} + \frac{B}{(1-2x)^2} + \frac{C}{1-5x}$$
Can anyone explain the intuition behind this? Is this a general rule that when you have a repeated factor in the denominator, you split it into all powers of that factor?

Yep - if there's a factor of $(x-a)^k$ in the denominator, you need to include $\frac{1}{x-a},\frac{1}{(x-a)^2},...,\frac{1}{(x-a)^k}$ terms in your expansion. 
Note that in your attempt, you could combine the first two terms to $\frac{A+B}{1-2x}$, making one of those constants redundant.
