Closed-form expression of generating function of this series

Question

I want to find closed-form expression of generating function of $<1,2,...,k,...>$. How can i figure that out?

Answer 1

Hint. Note that $$f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty}x^k.$$ That is $f$ is the generating function of the constant sequence $1,1,1,1,\dots$.

Answer 2

Try the following. First write down your generating function in its power series form. Then look for the closed form function that yields that power series as Maclaurin series expansion. For your particular problem I think the answer is $g(x) = \frac{x}{(1-x)^2}$.