This is a generalization of For $\alpha \in ]-1, 1[$, how to show that sequence $(u_{n + 1} - \alpha u_n)$ converges implies $(u_n)$ converges?
If $u_n-\sum_{k=1}^m c_ku_{n-k}$ converges, what conditions on the $c_k$ imply that $u_n$ converges?
Based on the source problem, my conjecture is that, if $f(x) =x^m-\sum_{k=1}^m c_kx^{m-k}$, then a sufficient condition is that the roots of $f(x)$, $(x_k)_{k=1}^m$ are all simple and satisfy $|x_k| < 1$.
I think a proof could be constructed by writing $d_n =u_n-\sum_{k=1}^m c_ku_{n-k}$, getting the expansion of $u_n$ in terms of the $x_k$ and the initial $u_j$, and then using $\lim_{n \to \infty} d_n $ exists and $|x_k|^n \to 0$ for all $k$.
Another way might be to get the generating function for $u_n$ in terms of $f(x)$.
However, I have other things to do, and this looks like it would take a while, so I'll leave it like this.