Let $f_n:[0,1] \to \mathbb R$ be defined by $\displaystyle f_n(x)= \frac{nx}{1+n^2 x^p}$ for $p>0$. Find the values of $p$ for which the sequence $f_n$ converges uniformly to the limit.
In previous excercise for $p=2$ I know that $f_n$ is not uniformly convergent. But I don't know how to proceed in this one. I think for $p<2$ and $p=2$ we can choose $x=1/n$ and show that it is not uniformly convergent. But i m not sure about this. I need a hint.
The point-wise limit is $0$. The sequence does not converge to $0$ uniformly for any $p>2$! To see this put $x=n^{-2/p}$. You get $f_n(x)=\frac 1 2 n^{1-2/p}$ and $1-2/p >0$.