Hmm, even as nicely as the accelerated solutions converge, it would seem that I'm only getting about 21-22 digits of accuracy from a 900-term solution (relative to a 1200-term solution, so granted, even that figure is only approximate).
Accuracy is better when we get close to integer tetrations. For \( {}^{\pi} e \), the 900- and 1200-term solutions agree to a little over 23 digits of accuracy. Therefore, I'm hopeful that the 1200-term solution is accurate to a few more digits than that, perhaps 25-27 digits.
So, to about 25 digits of accuracy, give or take, \( {}^{\pi} e \) is 37149801960.55698549872339920573987
Accuracy is better when we get close to integer tetrations. For \( {}^{\pi} e \), the 900- and 1200-term solutions agree to a little over 23 digits of accuracy. Therefore, I'm hopeful that the 1200-term solution is accurate to a few more digits than that, perhaps 25-27 digits.
So, to about 25 digits of accuracy, give or take, \( {}^{\pi} e \) is 37149801960.55698549872339920573987
~ Jay Daniel Fox