Let $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ and $\left( b _ { n } \right) _ { n \in \mathbb{N} }$ be two sequences of elements of $\mathbb{R}^{+*}$. We assume that $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ is decreasing and that $$\forall n \in \mathbb{N} , \quad \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k } = 1 .$$ We set, for $n \in \mathbb{N}$: $$B _ { n } = \sum _ { k = 0 } ^ { n } b _ { k } .$$ We assume in this question that there exists $C > 0$ such that $$b _ { n } \underset { n \rightarrow + \infty } { \sim } \frac { C } { n }$$ Using question 17 for a well-chosen sequence $\left( m _ { n } \right) _ { n \in \mathbb{N} }$, show that $$a _ { n } \underset { n \rightarrow + \infty } { \sim } \frac { 1 } { C \ln ( n ) }$$