We have $I _ { n } = \frac { p _ { n } + \zeta ( 2 ) q _ { n } } { d _ { n } ^ { 2 } }$ where $p _ { n }$ and $q _ { n }$ are non-zero integers for all $n \in \mathbb { N } ^ { * }$. Show that there exists $N \in \mathbb { N } ^ { * }$ such that for all $n \geqslant N$, $$0 < \left| p _ { n } + \zeta ( 2 ) q _ { n } \right| \leqslant \zeta ( 2 ) \left( \frac { 5 } { 6 } \right) ^ { n }$$ One may use, without proving it, the inequality $9 \frac { 5 \sqrt { 5 } - 11 } { 2 } \leqslant \frac { 5 } { 6 }$.
We have $I _ { n } = \frac { p _ { n } + \zeta ( 2 ) q _ { n } } { d _ { n } ^ { 2 } }$ where $p _ { n }$ and $q _ { n }$ are non-zero integers for all $n \in \mathbb { N } ^ { * }$.
Show that there exists $N \in \mathbb { N } ^ { * }$ such that for all $n \geqslant N$,
$$0 < \left| p _ { n } + \zeta ( 2 ) q _ { n } \right| \leqslant \zeta ( 2 ) \left( \frac { 5 } { 6 } \right) ^ { n }$$
One may use, without proving it, the inequality $9 \frac { 5 \sqrt { 5 } - 11 } { 2 } \leqslant \frac { 5 } { 6 }$.