Series with poles of Gamma function

Question

So I have a series: $\sum_{n=0}^\infty \frac{x^n}{\Gamma(-n)}$.

Can I just argue that $\lim_{n\to k} \frac{x^n}{\Gamma(-n)}$, $\forall k \in N_0$ is zero and thus the whole series becomes a null series?

Or do I have to justify it more? And in what way can I improve my argument?