I'm new to generating functions, and my lecture showed how to obtain exact closed form expression for Fibonacci numbers.
The coefficients of the generating function F(x) is the Fibonacci sequence {f_n}. After some manipulation,
$$\begin{align} (1-x-x^2)F(x) &= x \tag{A} \\ \\ F(x) &= \frac{x}{1-x-x^2} \tag{B} \\ \\ F(x) &= \frac{\frac{A}{1-a_1x}+\frac{B}{1-a_2x}}{\sqrt 5} \tag{C} \\ \\ F(x) &= \sum_{n=0} f_nx^n \tag{D} \end{align}$$
After doing the partial fraction decomposition, F(x) can then be written as a sum of 2 geometric series the coefficients of the series can be read off to obtain the closed form expression.
The equality in the first line is between formal power series; and the equality in the following lines is in terms of the function. I struggling to understand how the second line follows the first line.
Can someone please explain the sense of the steps?