$f(x)=1/(1+x^2)$. Lagrange polynomials do not always converge. why?

Let $f(x) = \frac{1}{1+x^2}$. Error of Interpolation with Lagrange polynomials for $n+1$ points is given by $$ e(x)=f(x)-P_n(x)=\frac{f^{(n+1)}(\eta_x)}{(n+1)!}\prod_{i=0}^n (x-x_i) $$

Carl Runge said that for this $f$, Lagrange polynomial $P_n$ do not converge uniformly. How can I show this?

Here is an elementary proof that equispaced Lagrange polynomials on $[-a,a]$ do not converge at $b$ for $a=5$ and $b=4$ (and many other values). We assume an odd number of nodes $2n+1$, so the nodes are at $ak/n$ for $-n \leq k \leq n$; an even number of nodes is similar but notationally a little uglier. Let $p(x)$ be the Lagrange interpolating polynomial. Let $q(x) = \prod_{k=-n}^n (x-ak/n)$, the monic polynomial vanishing at the nodes.

So $p(x)(x^2+1)-1$ vanishes at the nodes, and we thus have $$p(x) (x^2+1)-1 = q(x) r(x) \quad (\ast)$$ for some polynomial $r(x)$. The polynomial $p$ has degree $\leq 2n$ and $q$ has degree $2n+1$, so $\deg r \leq 1$. Also, note that $p(x)$ is an even polynomial (obtained by interpolating an even function at symmetrically placed points) and $q(x)$ is an odd polynomial (a root at $0$, and all other roots mirror symmetric) so $r$ is odd. Thus, $r(x) = c x$ for some constant $c$.

We can compute $c$ by plugging in $x=i$. (If I were teaching this to students who fear complex numbers, I might first write $p(x) = f(x^2)$, $q(x) = x g(x^2)$, set $y=x^2$ and plug in $y=-1$. But here we will be fearless!) $$p(i) \cdot 0 -1 = (c i) \prod_{k=-n}^n (i-ak/n) = (ci) \cdot i \cdot \prod_{k=1}^n (i^2 - a^2 k^2/n^2)$$ $$c =\frac{(-1)^n}{\prod_{k=1}^n (1+a^2 k^2/n^2)}.$$

Rearranging equation $(\ast)$ gives an exact formula for the error: $$p(x) - \frac{1}{1+x^2} = \frac{c x q(x)}{1+x^2} = \frac{(-1)^n x^2 \prod_{k=1}^n (x^2-a^2 k^2/n^2)}{(1+x^2) \prod_{k=1}^n (1+a^2 k^2/n^2)}.$$

We see that the error goes to zero if and only if $$\lim_{n \to \infty} \left( \sum_{k=1}^n \log {\Large |} x^2 - a^2 k^2/n^2 {\Large |} - \sum_{k=1}^n \log ( 1+a^2 k^2/n^2 ) \right) = - \infty.$$

The sums look like Riemann sums with spacing $a/n$, so the quantity in the limit is approximately $$\frac{n}{a} \cdot \left( \int_{t=-a}^a \log| x^2 - t^2| dt - \int_{t=-a}^a \log( 1+ t^2) dt \right).$$ More precisely, we can justify turning sums into integrals as long as $x$ isn't too close to one of the nodes: For example, if we take a sequence of points $x_n$ approaching $x \in [-a,a]$, with $x_n$ always halfway between two nodes, this is valid.

So the limit will be $\pm \infty$ according to the sign of $$\int_{t=-a}^a \log |x^2 - t^2| dt - \int_{t=-a}^a \log (1+t^2) dt.$$ The integrals can be done in closed form, but I'll close by just plotting the graph for $a=5$:

As you can see, the curve crosses the axis around $3.8$, which is exactly where the Lagrange polynomials stop converging.

Any textbook answers this question by referring to Runge's work, which is dated, and not translated in English.

Not true. The textbook Numerical analysis by Kinkaid and Cheney refers to the article On the Runge example by James F. Epperson, published in: American Mathematical Monthly Volume 94 Issue 4, April 1987; pages 329-341.

As the length of the article suggests, detailed treatment is a bit too long to reproduce here. Besides, the article is freely available from MAA website. (MAA publishes the journal, so I'm assuming this is a legitimate copy.)

As for the original question:

Carl Runge said that for this $f$, Lagrange polynomial $P_n$ do not converge uniformly.

This is rather vague. Lagrange polynomials do converge if the nodes are chosen well. Also, even with equidistant nodes they converge when the interval of interpolation is not too large. If you wanted to discuss a particular interval and a set of nodes, you should include it in the question. But read the aforementioned article first.