# Inspired by Archit Boobna

Calculus Level 4

For every real number $$z$$, define the infinite nested logarithm as the limit of the series

$a_1 = \ln z \\ a_ 2 = \ln_{ \ln z } z \\ a_3 = \ln_{ \ln _{ \ln z } z } z \\ \vdots$

where $$a_{n+1} = \ln_{a_n} z$$.

For what value of $$Z$$, does the limit of the sequence $$a_n$$ exist for all $$z > Z$$?

Inspiration, see solution comment

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