Let $f_n(x)=n^2 x(1-x^2)^n$ ($0 \le x \le 1, n=1,2,3...$)
For $0
Theorem: If $p>0$ and $\alpha$ is real, then $\lim_{nā\infty}\frac{n^\alpha}{(1+p)^n}=0$
I have no idea how to apply this theorem to this limit value.
I think it's impossible to apply that theorem, becaus in here,
if we put $n=-n, \alpha=2, p=-x^2$ then $-x^2<0$.