how to proof this inequality or give a more accurate boundary？ $$ 1 + x + \frac{{{x^2}}}{{2!}} + ....... + \frac{{{x^n}}}{{n!}} > \frac{{{e^x}}}{2}，x \in [0,n]\ $$

I meet anothor problem $$ \mathop {\lim }\limits_{n \to \infty } \frac{{1 + n{\rm{ + }}\frac{{{n^2}}}{{2!}} + ..... + \frac{{{n^n}}}{{n!}}}}{{{e^n}}} = \frac{1}{2} $$ maybe I can use this this inequality to solve this problem

How can we prove this inequality or give a more accurate bound? $$ 1 + x + \frac{{{x^2}}}{{2!}} + ....... + \frac{{{x^n}}}{{n!}} > \frac{{{e^x}}}{2}，x \in [0,n]\ $$

I came across the equation: $$ \mathop {\lim }\limits_{n \to \infty } \frac{{1 + n{\rm{ + }}\frac{{{n^2}}}{{2!}} + ..... + \frac{{{n^n}}}{{n!}}}}{{{e^n}}} = \frac{1}{2} $$ Maybe I can use this this inequality to solve this problem