Simple Question about e^(1/e)

[rsgerard]

Thanks very much for your reference. I'm still doing a lot of research on this. This is extremely helpful, I'll follow up in the future.

Ryan

By some incidence I just found the article:
Barrow, D. F. (1936). Infinite exponentials. Amer. math. Monthly, 43, 150–160.

He states there in Theorem 7:
The infinite exponential \( E_{i=0}^\infty (e^{1/e}+\epsilon_i) \), where \( \epsilon_i \) are all positive or 0,
(a) will converge if \( \lim_{n\to\infty} \epsilon_n n^2 < \frac{e^{1/e}}{2e} \)
(b) will diverge if \( \lim_{n\to\infty} \epsilon_n n^2 > \frac{e^{1/e}}{2e} \).

However the numbering is kinda reverse to yours.
\( E_{i=0}^n a_i \) means \( a_0\^a_1\dots \^a_n \).
I.e. the tower will be enlarged on top (infinity on top), while your tower gets enlarged on the bottom (infinity at bottom).
But anyway its good to know that people dealt already with such questions.
And perhaps reading this article helps you with your problem.