Find points $x$ such that $p_n(x)-f(x)$ does not converge to $0$ as $n\to\infty$

Question

Let $f(x)=\frac{1}{1+25x^2}$ and range is $[-1,1]$. Given $n+1$ equidistant points $x_0 = -1,x_1,...,x_n = 1$ and their values $f(x_0),f(x_1),..,f(x_n)$, perform polynomial interpolation by the $n+1$ points and get the polynomial $p_n(x)$. The question is

Find $x\in[-1,1]$ such that error $p_n(x)-f(x)$ does not converge to $0$ as $n\to\infty$.

My experiments on matlab shows it looks like that there is no such point where the error goes to $0$ as $n\to \infty$, but I want to know if there is any analytical proof of this. Thank you!

Answer 1

The phenomenon you want to find by numerical simulation,

is called the Runge's phenomenon

wiki:

https://en.wikipedia.org/wiki/Runge%27s_phenomenon

you can get details of it.