Closed Form Generating Function for sum of natural numbers I need to find a Closed Form Generating Function for a sequence whose $n$-th term is the sum of the first $n$ natural numbers, i.e:
$$f(x) = \sum_{i=1}^{n}\frac{n(n+1)}{2}x^n$$
and am having difficulties. I am trying to approach it through differentiation since I could not find a way to do it recursively but I am not sure which apporach is right.

Ok I have it figured out.
The answer is : $$f(x) = \frac{x}{(1-x)^3}$$
We get this by starting with $\frac{1}{1-x} = 1 + x + x^2 + ...$, taking the second derivative yields $2 + 6x + 12x^2 + ... = \frac{2}{(1-x)^3}$ then multiplying by $x\over2$ gives the desired series.
