I want to approximate $1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!}$ for finite $n$, where $n \geq x > 0$ and $n \in \mathbb{N}$.
I am wondering if there exits tight upper and lower bounds $U(x, n)$ and $L(x, n)$ such that
$$L(x, n) \leq \sum\limits_{i=0}^n \frac{x^i}{i!} \leq U(x, n).$$
A well-known result is $\lim_{n \to \infty} \sum\limits_{i=0}^n \frac{x^i}{i!} = e^{x}$, but how about finite $n$?