05/26/2010, 04:55 PM
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.
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.