They tell that if $f(X)=\sum_{k}\alpha X^{k}$ is a power matrix series then use the Cayley-Hamilton theorem to rewrite formally $f(X)$ as : $$ f(X) = c_{0}(X)+ c_{1}(X)X+\cdots+c_{n-1}(X)X^{n-1}$$ Where $c_{j}(X)$ are (multiple) power series in the matrix entries of $X$ which are invariant under conjugation (i.e. $c_{j}(aXa^{-1})=c_{j}$(X) ) and 'formally' means that I don't have to worry about convergence when rearranging the series.
I feel completely lost about this exercise, I would appreciate a hint or a solution of course.
Hint: The Cayley–Hamilton theorem implies that $X^n=g(X)$ for some polynomial $g$ of degree less than $n$. Therefore, every power $X^k$ can be reduced to a polynomial $g_k(X)$ of degree less than $n$.