I Smell Taylor Series

For any x it is true that:
$e^{x}=1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+\frac{x^{5}}{5!}...$
So for any $x\neq 0$ :
$e^{x}\neq 1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}$
But for $x=0$ all the extra terms become $0$ so the equation holds true. Therefore, the sum of the roots is 0.

Note: I feel like i explained it poorly or maybe i've made a mistake. Please provide a better solution.