Can someone please help me with this proof?

Hi. In one of my classes, I have to prove the following theorem. Given that A is an nxn matrix, and and x(t) is a vector in R-n, prove the following:

If x'(t) = A*x(t), where x(0) = x-not,
then x(t) = (x-not)*e^(A*t)

The first step is for me to prove that the following series converges:

(the sum as k goes from 0 to infinity of)((t^k)(A^k)/k!)

I also need to use the Frechet Derivative, linear systems of differential equations, and the relation to the fundamental matrix.

If someone could help me get started, I'd be very grateful. Thanks!