Can I use the Polynomial Interpolation Error Theorem to conclude that $p_n(x)$ tends to $f(x)$ as $n$ goes to $\infty$?

Question

I was doing some revision questions on the polynomial interpolation error theorem, $$f(x)-p_n(x) = \frac{f^{n+1}\pi_{n+1}(x)}{(n+1)!}$$ After doing out the proof of this theorem, I'm asked if you can use this theorem to conclude that $p_n(x)$ tends to $f(x)$ as $n$ goes to $\infty$?

My initial thoughts are that it can be used as $(n+1)!$ will go to infinity meaning $\frac{f^{n+1}\pi_{n+1}(x)}{(n+1)!}$ will go to zero but I'm not sure I'm right. Anyone able to tell me if I'm wrong?

Answer 1

You cannot conclude that, the sequence of Lagrange polynomials of $f$ does not always converge pointwise towards $f$, the reason being that $\|f^{n+1}\|_{\infty}$ may explode faster that $(n+1)!$. This is known as the Runge's phenomenon.

By the way, the error term at $x$ of $f\colon]a,b[\rightarrow\mathbb{R}$ is given by: $$\frac{f^{n+1}(\xi_x)\pi_{n+1}(x)}{(n+1)!},$$ where $\xi_x\in]a,b[$, you forgot to write $\xi_x$.