Let $$x_{n+1} = a_n x_n + b_n$$ a recursion relation so that $x_n \rightarrow x$.
Let $$y_{n+1} = a_n y_n + b_n + c_n$$ a perturbed relation.
I am looking for sufficient conditions on $c_n$ so that $y_n \rightarrow x$.
Is there a textbook/reference?
Thank you