I'd like your help with the deciding whether the following function uniformly converges in two intervals.
$f_n(x)=n \left(x^{\frac{1}{n}}-1 \right) .$
In $(0,10]$, with l'Hôpital, I showed that for every $x$ the limit of the function is $\ln x$, so with a help from Dini's theorem, I am able that the function is monotone and the limit is continuous so the function uniformly converges.
I'm not sure what should I do with the other interval. I tried to evaluate the limit of $|f_n(x)-f(x)|$, I didn't come to any smart conclusion.