PROPOSITION IX.A.1: If $P_n(x)= 1 + x+ {x^2}/2 + {x^3}/{2*3} + {x^4}/{4!}+...+{x^n}/{n!}$ then
$P_n(x)$ is a polynomial of degree n so that $P_n(0)=
P_n'(0)=P_n''(0)=...=P_n^{(n)}(0)=1$ and $P_n(b)$ is approximately equal
to $e^b$.
In fact, if we let $R_n = e^b - P_n(b)$, then for some $c$ between $0$ and $b$, $R_n = e^c {b^{n+1}}/{(n+1)!}$ .
GeoGebra Graph and Mapping Diagram of $P_n(x)$ and $R_n(x)$