Infinite Exponentation: Convergence Tests and More

[Jakim]

Infinite exponentiation

Define:

$\displaystyle\bigodot\limits_{i=1}^n a_i = {a_1}^{{a_2}^{{...}^{a_n}}}=\exp(\ln a_1 \cdot \exp(\ln a_2 \cdot \exp(\ln a_3 \cdots (\exp(\ln a_{n-1} \cdot a_n))))$

Has anybody thought about convergence test for real numbers for:

$\displaystyle\bigodot\limits_{i=1}^{\infty} a_i = \lim_{n\to\infty}\bigodot\limits_{i=1}^n a_i$

We should also add note that $\bigodot\limits_{i=1}^{\infty} a_i = 0$ diverges to $0$.

My shot is: the infinite exponentation converges if and only if contains only finite amount of terms $|a_i| \geqslant e^{e^{-1}} \vee |a_i| \leqslant \left(e^{-1}\right)^e$ but I haven't much thought about it.

Greetings.

 

[Mandlebra]

Check this article out http://en.wikipedia.org/wiki/Tetration

You should be pleasantly surprised :)