Using the Weierstrass M-test I can show that the function series $$\sum_{n=1}^{\infty}\frac{n^x}{3^n-5}$$ is uniformly convergent on any closed bounded interval $[0,a]$.
I have a feeling that the series is not uniformly convergent for $x \in [0, \infty)$ since the numerator can not be bounded but how can I show that formally?