Infinite Pentation (and x-srt-x)

[bo198214]

If we interpolate these points, one would except the interpolation to diverge for infinite pentation, and x-superroot-x, but the interpolating polynomial of these integer points do seem to converge,

Now I doubt about the interpolation method.
If the interpolation of the self-tetra-root would yield a valid function,
then shoud the interpolation of the simple self-root \( n^{1/n} \) also converge to the self-root \( x^{1/x} \) on \( x>0 \).
But this seems not to be the case.

An interpolation polynomial of degree 400 (401 sample points) still has a negative value at 0.25.
   
And if we compare the values it seems that the negativity gets rather worse:
101 points: \( f(0.25)\approx -0.235 \)
201 points: \( f(0.25)\approx -0.330 \)
301 points: \( f(0.25)\approx -0.364 \)
401 points: \( f(0.25)\approx -0.378 \)

So it really looks as if the interpolation (even if it converges) does not converge to \( x^{1/x} \) which is positive everywhere.
So I would conclude that the interpolation of the self-tetra-root also does not converge to a self-tetra-root, even if it converges.