Generating function for $a_n=n$, $n\ge0$

Question

Given the sequence $a_n$ where $a_n=n$ in every $n\ge0$. Find the generating function of $a_n$.

Of course this means the generating function is $\sum_{n=0}^\infty nx^n$. But how to simplify?

Answer 1

Hint: Consider $g'(x)$ for $g(x)=1+x+x^2+\cdots$.

Answer 2

Note that

$$ \frac{1}{(1-x)^2}=\left(\sum_{n=0}^\infty x^n\right)\left(\sum_{n=0}^\infty x^n\right) =\sum_{n=0}^\infty\left(\sum_{k=0}^n 1\right)x^n =\sum_{n=0}^\infty (n+1)x^n. $$

Hence $$ \frac{x}{(1-x)^2}=\sum_{n=0}^\infty nx^n. $$