# How could I show the following "fact" about exponential series?

Could you please give me some hint on how to show (if possible) that the following inequality holds?

$$\bigg|(-1)^m\frac{t^m x^m}{m!}+\sum_{j=m+1}^{\infty}\frac{(-1)^jx^jt^j}{j!}\bigg|\leq \bigg|(-1)^m\frac{t^m x^m}{m!}\bigg|$$

(where $$x>0$$)

notice that the left side is nothing more than

$$\bigg|e^{-tx}-\sum_{k=0}^{m-1}\frac{(-1)^jx^jt^j}{j!}\bigg|$$

Any hint or advice will be really appreciated.

For $$y\geqslant 0$$ we have $$1\geqslant e^{-y}=\sum_{k=0}^{\infty}(-1)^k y^k/k!$$; integrating this $$m$$ times, we get $$\frac{y^m}{m!}\geqslant\sum_{k=0}^{\infty}\frac{(-1)^k y^{k+m}}{(k+m)!}\underset{k+m=j}{=}(-1)^m\sum_{j=m}^{\infty}(-1)^j\frac{y^j}{j!}.$$