Construct a sequence of polynomials that converge uniformly on $[0,1]$, but whose derivatives fail to converge uniformly.

Question

Can anyone help me with this? In what situation would the derivative of a polynomial not converge uniformly given the polynomial itself converges uniformly on a compact set.

I don't even know where to start. I was about to use Weierstrass Approximation theorem, but I don't know how to use it.

Answer 1

$f_n(x) = x^n$ is a standard example where the sequence does not converge uniformly on $[0,1]$. So trying

$$g_n(x) = \frac{1}{n+1} x^{n+1} = \int_0^x t^n dt$$

and this actually satisfies your conditions.