Suppose that I have two matrices $A$ and $B$ and I know that $A^n\to B$ as $n\to \infty$. Clearly, when we pick some norm one will find that $\|A^n-B\|\to 0$ as $n\to \infty$.
Now let’s say that I know $\|A^n-B\|=f(n)$ where $f(n)$ tends to zero for large $n$. How would one express $A^n$ as a function of the matrix $B$ and some error involving the function $f(n)$?
I thought we could write $$A^n = B+f(n)\,C$$ where $C$ is some matrix of norm $1$. But what should this matrix be? Does it matter since $f(n)\to 0$ with $n\to\infty$?