Let $A$ be a $m\times m$ complex matrix.
In my text, it proves the convergence of $\sum \frac{A^n}{n!}$ by using Jordan canonical form which is quite tricky.
However, isn't it much easier to prove the convergence of the summation in the following way?
Note that $||\frac{A^n}{n!}||≦\frac{||A||^n}{n!}$. Since the summation of the right hand side converges and $M_n(\mathbb{C})$ is finite-dimensional, hence a Banach space, $\sum \frac{A^n}{n!}$ is convergent.
Is my argument wrong?
If I'm correct, I cannot understand why this text proves the convergence using a missile to break a stone, namely Jordan canonical form..
Your argument is excellent.
The books use Jordan form because when studying the properties of the matrix exponential, Jordan form is very useful; and as long as they are going to be introducing the "missile" they might as well use it for the easier part of proving convergence.