Generating functions for the non negative integer $k$-th powers

Question

Prove that for all $k \in \mathbb{N}\cup \{0\}$, the generating function for the non negative integer $k$-th powers is a quotient of polynomials in $x$, that is for all $k \in \mathbb{N} \cup \{0\}$ there are polynomials $R_k(x)$ $S_k(x)$ such that the coefficient of $x^n$ in $\frac{R_k(x)}{S_k(x)}$ is $n^k$.

Answer 1

Let $F_k(x)=\sum_{n\geq 0}n^k x^n$ be the the generating function for the $k$-th powers.

Then we have that:

1) $F_0(x)=\frac{1}{1-x}$,$\qquad$ 2) For $k\geq 0$, $F_{k+1}(x)=x\cdot \frac{d}{dx}\left (F_{k}(x)\right)$.

Since the derivative of a rational function is a rational function, by induction we may conclude that each $F_k$ is a rational function.