Does anyone have proof that is worth (or not) or suggestion to this?
Let be $S_n \in \sum_{k=1}^n \frac{1}{k^2}$ ($\zeta(2)$ approximation), then
$$ \left|S_n - \frac{\pi^2}{6} \right| < \frac{1}{n} $$
and
$$ \left|S_{n-1} - \frac{\pi^2}{6} \right| > \frac{1}{n} $$