The question is:
- Find the ordinary power series generating functions of each of the following sequences, in simple, closed form. In each case the sequence is defined for all $n\geq0$. (a) $a_n=n$
This is the function $A(x)=0x^0+1x^1+2x^2+3x^3+\ldots$, which I rewrite as: $A(x)=(x+x^2+x^3+x^4+x^5+\ldots)+(x^2+x^3+x^5+\ldots)+(x^3+x^4+x^5+\ldots)+\ldots$
Each of the terms is a geometric series, and so I wrote this as
$\begin{align} A(x) &= \frac{x}{1-x}+\frac{x^2}{1-x}+\frac{x^3}{1-x}+\ldots \\ &= \frac{x}{1-x}\left(1+x+x^2+\ldots\right) \\ &=\frac{x}{(1-x)^2} \end{align}$
Therefore, my answer would be $A(x)=\frac{x}{(1-x)^2}$. However, the answer key at the back of the book says that the answer is "$(xD)(1/(1-x))=x/(1-x)^2$". While my generating function looks like the RHS of the answer key, I don't understand what the LHS means. Is there something I'm missing here?
The $LHS$ is simply another way to get the same generating function that you found. It says: take the derivative of $1/(1-x)$ and then multiply by $x$.