I'm trying to prove the smoothness of exponential map for complex matrix as a map from $C^n\to C^n$. In particular, given $A\in M_n(C)$ we have $e^A=\sum_n\frac{1}{n!}A^n$. The easiest solution I've read says:
$A^n$ has component which are $n$-homogenous polynomial in the components of A. Therefore, $e^A$ has component which are convergent power series (in the component of A) in $M_n(C)$.
For me, it is not clear how to find that power series for the components. For example, if $A=(a_{ij})$, then the $ij$-component of $A^n$ is $\sum_{k_1, \dots, k_{n-1}=1} a_{i,k_1} a_{k_1,k_2}... a_{k_{n-1},j}$ (the $n$-homogenous polynomial). When I fix the components of A (except one, say x) I can't write $\sum_{n=0}^m \frac{1}{n!}A^n=\sum_{n=0}^m a_nx^n$ for some fixed scalar $a_n$ because $a_n$ would change for every $m$.