I study different cases about $a \in R$ :
If $a=0, f_n(x)=n(1-x^2)^{n}$. If $|x|\le1$ the $\lim_{n\rightarrow +\infty} f_n =0$, in the other case it doesn't esists. If $a>0$? $\lim_{n\rightarrow +\infty}n^{a}x(1-x^2)^{n}=0$ for $|x|\le 1$ and not esists for $|x|\ge1$. Idem per $a<0$