I'm having trouble knowing where to start with this assignment.
Let $A = \{a_{ij}\}_{i,j=1,\dots,n}$ be a matrix, and let $\eta = max_i(-a_{ii})>0$.
With $P=I+\frac{1}{\eta}A$, where $I$ is the identity matrix.
Let $‖\cdot‖$ be the matrix norm induced by the maximum norm for $\mathbb{R}^n$. Show that
$$‖\exp(Ax)-e^{-\eta x}\sum_{n=0}^l\frac{(\eta x)^n}{n!}P^n‖<\epsilon$$
if
$$\sum_{n=0}^l\frac{(\eta x)^n}{n!}e^{-\eta x}>1-\epsilon$$
I've shown that
$$\exp(Ax)=-e^{-\eta x}\sum_{n=0}^{\infty}\frac{(\eta x)^n}{n!}P^n$$