In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Deduce that there exists a sequence $\left( C _ { k } \right) _ { k \in \mathbf { N } }$ of strictly positive reals, independent of $p$, such that
$$\forall k \in \mathbf { N } , \left| \mathbf { E } \left( X ^ { k } \right) - \frac { 1 } { p ^ { k } } \right| \leq \frac { C _ { k } q } { p ^ { k } }$$