Is the sequence of functions $f_n(x)=n^\alpha x(1-x^2)^n$ uniformly convergent on the interval $[0,1]$ for $0<\alpha<\frac{1}{2}$? It converges pointwise to $f=\underline{0}$, since for $x=0$ or $x=1$, $f_n(x)=0, \forall n\in\mathbb{N}$ and for $x\in [0,1]$ we have $ 0<f_n(x), \forall n\in\mathbb{N}$ and
$$ \begin{align} \lim_{n\to\infty} \frac{f_{n+1}(x)}{f_n(x)}=1-x^2<1 \end{align} $$
I don't know if there are any $\alpha\in [0,\frac{1}{2}]$ which make this convergence uniform. I'd say it is the whole interval, but I don't know how to show it.