upper bound of exponential function I am looking for a tight upper bound of exponential function (or sum of exponential functions):
$e^x<f(x)\;$ when $ \;x<0$ 
or
$\displaystyle\sum_{i=1}^n e^{x_i} < g(x_1,...,x_n)\;$ when $\;x_i<0$ 
Thanks a lot!

For $x < 0$ and $n \in \mathbb{N}$,
$$
0 < \sum_{k=0}^{n} \frac{(-x)^k}{k!} < e^{-x}
$$
so
$$
e^x < 1\left/\sum_{k=0}^{n} \frac{(-x)^k}{k!}\right..
$$
