Uniform convergence of geometric series $\sum_{n=1}^{\infty} e^n x^n e^{-nx}$

Question

I am trying to prove that the series $\sum_{n=1}^{\infty} e^n x^n e^{-nx} $ converges uniformly on $[0,a]\bigcup[b,\infty)$ where $a<1, b>1$ is it enough to prove uniform convergence on the two intervals separately? Can this be done with the Weierstrass M test ?