I'm trying to find the intervals where $\sum n^x x^n$ is uniformly convergent.
I was able to get that it's not uniformly convergent on $x \geq 1$ and that it's uniformly convergent on $-1 \leq x < 1$. Am I correct for these?
How do I work it out for $x < -1$? I was thinking of using comparison test but I'm not sure if I'm in the right direction.
When $x<-1$, we put $a_n=n^x x^n, n\in \mathbb{N}.$
We have $\sqrt[n]{|a_n|}= \frac{|x|}{\sqrt[n]{n^{-x}}}.$ So, $$\lim \sqrt[n]{|a_n|}=|x|>1$$. Therefore, the series diverges.