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

[bo198214]

Now calculate the polynomial sum of \( \left( 1 + \frac{x^2}{n} \right)^n \) (using Faulhaber's formula).

This seems to convergent to a nice function.

But wait, first it *seems* to converge, however later it takes off. E.g. if I compute the coefficient of x, I get rounded to two significant digits (scroll down):

Code:

n= 60 : 1.2
n= 61 : 1.2
n= 62 : 1.2
n= 63 : 1.1
n= 64 : 1.2
n= 65 : 1.1
n= 66 : 1.2
n= 67 : 0.96
n= 68 : 1.7
n= 69 : -0.14
n= 70 : 4.5
n= 71 : -7.8
n= 72 : 25.
n= 73 : -64.
n= 74 : 180.
n= 75 : -500.
n= 76 : 1400.
n= 77 : -4100.
n= 78 : 12000.
n= 79 : -35000.

This is not a rounding error as I compute with fractions and do the rounding only for better visibility.