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+--- Thread: Infinite Pentation (and x-srt-x) (/showthread.php?tid=269)
RE: Infinite Pentation (and x-srt-x) - bo198214 - 05/31/2011
(04/11/2009, 09:16 AM)andydude Wrote: 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.