Transseries, nest-series, and other exotic series representations for tetration

[bo198214]

Mike, I have another bad news, I found out that not even the Faulhaber sum of \( e^{\frac{x^2}{2}} \) is convergent, contrary to what I said before because it looks convergent up to n=100 terms; so your transseries sum \( e^{e^x} \) can not be convergent.

The first coefficient of the Faulhaber sum for \( n \) terms:

Code:

n= 100 : 1.1
n= 101 : 1.0
n= 102 : 1.0
n= 103 : 1.2
n= 104 : 1.2
n= 105 : 0.68
n= 106 : 0.68
n= 107 : 2.2
n= 108 : 2.2
n= 109 : -1.8
n= 110 : -1.8
n= 111 : 9.2
n= 112 : 9.2
n= 113 : -22.
n= 114 : -22.
n= 115 : 67.
n= 116 : 67.
n= 117 : -190.
n= 118 : -190.
n= 119 : 570.