10. In the general case, show that $( u _ { n } )$ converges and specify its limit.
Part 3: Linear recurrent sequences with variable coefficients
We keep the notation from the previous part and we now consider a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ satisfying a recurrence of the form $$v _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } b _ { i } ( n ) v _ { n + i }$$ where $v _ { 0 } , \ldots , v _ { d - 1 }$ are given and for all $i \in \{ 0 , \ldots , d - 1 \}$, $\left( b _ { i } ( n ) \right) _ { n \geqslant 0 }$ is a sequence with complex values converging to $a _ { i }$. We also define for all $n \geqslant 0$, $V _ { n } = \left( v _ { n } , \ldots , v _ { n + d - 1 } \right)$. We always assume hypothesis (*) is satisfied.
10. In the general case, show that $( u _ { n } )$ converges and specify its limit.
\section*{Part 3: Linear recurrent sequences with variable coefficients}
We keep the notation from the previous part and we now consider a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ satisfying a recurrence of the form
$$v _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } b _ { i } ( n ) v _ { n + i }$$
where $v _ { 0 } , \ldots , v _ { d - 1 }$ are given and for all $i \in \{ 0 , \ldots , d - 1 \}$, $\left( b _ { i } ( n ) \right) _ { n \geqslant 0 }$ is a sequence with complex values converging to $a _ { i }$. We also define for all $n \geqslant 0$, $V _ { n } = \left( v _ { n } , \ldots , v _ { n + d - 1 } \right)$. We always assume hypothesis (*) is satisfied.\\