Find the limit function $f$ of $\{F_n\}$. $F_n=nx^n(1-x^2) $
My solution:
For $x=\pm1, 0, F_n=0 \implies f(0)=f(1)=f(-1)=0$.
For $|x|>0$. we have $$(1-x^2)\lim_{n\to\infty} nx^n=\infty\text{ , if }x>1$$
and $$(1-x^2)\lim_{n\to\infty} nx^n=-1^n\infty\text{ if }x<-1$$
Finally, for $0<|x|< 1,$ we have $$(1-x^2)\lim_{n\to\infty} nx^n= (1-x^2)\lim_{n\to\infty} \dfrac{1}{-nx^{-n-1}}== -(1-x^2)\lim_{n\to\infty} \dfrac{x^{n+1}}{n}=0$$
Hence {F_n} converges pointwise to $f=0$ on the interval $[-1,1]$