Let $n \in \mathbb { N } ^ { * }$. Justify that $\zeta ( 2 ) = \sum _ { k = 1 } ^ { + \infty } \frac { 1 } { k ^ { 2 } }$ is well-defined, then show that we can write
$$\sum _ { k = 1 } ^ { n } \frac { 1 } { k ^ { 2 } } = \frac { p _ { n } } { q _ { n } }$$
with $p _ { n } \in \mathbb { N } ^ { * }$ and $q _ { n } = d _ { n } ^ { 2 }$.