For any positive integer $n$, we define $$S_n=\sum_{k=1}^n \frac{2^k}{k^2}.$$Prove that there are no polynomials $P,Q$ with real coefficients such that for any positive integer $n$, we have $\frac{S_{n+1}}{S_n}=\frac{P(n)}{Q(n)}$.
Source: 2021 Peru TST D1P3
Tags: number theory, algebra
For any positive integer $n$, we define $$S_n=\sum_{k=1}^n \frac{2^k}{k^2}.$$Prove that there are no polynomials $P,Q$ with real coefficients such that for any positive integer $n$, we have $\frac{S_{n+1}}{S_n}=\frac{P(n)}{Q(n)}$.