Proof of Convergence of Matrix Exponential Series

[rayman123]

Homework Statement

I am searching for a proof of convergence of matrix exponential series. Where I can find it?
Thanks!

Homework Equations

The Attempt at a Solution

 

[micromass]

Can you show that the series converges absolutely? I.e. that

$$\sum_{k=0}^{+\infty}{\frac{\|A^k\|}{k!}}$$

converges...

 

[SVXX]

@Micromass : As a sidenote, can this expression be checked for convergence/divergence using D'Alembert's Ratio test?

 

[micromass]

Probably yes, but I prefer to compare it with the series expansion of $$e^{\|A\|}$$...

 

[rayman123]

I found the whole proof