As we all know, there is a thing about polynomial interpolation when using equally-spaced nodes on certain functions called Runge's phenomenon. However, there are a lot of functions that do not have this problem (those whose radius of absolute convergence is infinite).
Given the function $f(x) ={}^x2$, meaning $2$ tetrated to $x$, where tetration means the hyperoperation of the exponential:
$${}^42 = 2[4]4 = 2\upuparrows4 = 2^{2^{2^{2}}} = 2^{2^{4}} = 2^{16} = 65536$$
We know its values for integers $x = k \geq-1$. Therefore, we should be able to construct an infinite Lagrange interpolating "polynomial" as follows:
$$p_\infty(x) = \sum_{j=-1}^{\infty}{}^{j}2 \prod_{\substack{i=-1\\i\neq j}}^{\infty} \frac{x-i}{j-i}$$
The question is simple: does this series converge uniformly/absolutely for all $x\in(-2,+\infty)$? If not, which is the largest interval of uniform/absolute convergence?