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Infinite exponentiation
Define:
[itex]\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))))[/itex]
Has anybody thought about convergence test for real numbers for:
[itex]\displaystyle\bigodot\limits_{i=1}^{\infty} a_i = \lim_{n\to\infty}\bigodot\limits_{i=1}^n a_i[/itex]
We should also add note that [itex]\bigodot\limits_{i=1}^{\infty} a_i = 0[/itex] diverges to [itex]0[/itex].
My shot is: the infinite exponentation converges if and only if contains only finite amount of terms [itex]|a_i| \geqslant e^{e^{-1}} \vee |a_i| \leqslant \left(e^{-1}\right)^e[/itex] but I haven't much thought about it.
Greetings.
Define:
[itex]\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))))[/itex]
Has anybody thought about convergence test for real numbers for:
[itex]\displaystyle\bigodot\limits_{i=1}^{\infty} a_i = \lim_{n\to\infty}\bigodot\limits_{i=1}^n a_i[/itex]
We should also add note that [itex]\bigodot\limits_{i=1}^{\infty} a_i = 0[/itex] diverges to [itex]0[/itex].
My shot is: the infinite exponentation converges if and only if contains only finite amount of terms [itex]|a_i| \geqslant e^{e^{-1}} \vee |a_i| \leqslant \left(e^{-1}\right)^e[/itex] but I haven't much thought about it.
Greetings.
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