If $S(n)=\sum_{i=1}^{n} u_i \in \Bbb Q$ for every $n$, do we have $S(n) = P(n)/Q(n)$ for some $P, Q\in \mathbb{Q}[x]$?

Question

Denote $S(n)=\sum_{i=1}^{n} u_i$. I have a question that if $S(n)\in \mathbb{Q}, \forall n \in \mathbb{N}$, can I prove that $S(n)=P(n)/Q(n)$ with $P(x), Q(x)\in \mathbb{Q}[x]$, or can someone show me a counter example for that?

Answer 1

Let $u_n=10^n$. Then $S_n ≥ 10^n$ and for any non-zero polynomials $P,Q \in \Bbb Q[X]$, we have $$\dfrac{S_n}{|P(n)/Q(n)|}=\dfrac{|Q(n)| S_n}{|P(n)|} ≥ \dfrac{a \cdot 10^n}{|P(n)|} \to \infty$$

as $n \to\infty$ (since $a := \min\limits_{n \in \Bbb N} |Q(n)| > 0$, otherwise $P(n)/Q(n)$ is not well-defined).

So in particular, we can't have $S_n = P(n)/Q(n)$ for all $n$.

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However, we can always find a polynomial $P$ with rational coefficients such that $P(n)=S_n$ where $0≤n≤N$, for any $N≥0$.