Is there a reduction for this infinite sum?
$$\log x + \log\log x + \log\log\log x +... = ?$$
for all $x > 0$?
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2Possible duplicate of In which interval (domain) does the $\sum_{n=1}^{\infty}\log^n(1+x)$ converge absolutely? – Tito Eliatron Jan 24 '19 at 20:47
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The sum diverges with any reasonable interpretation of the logarithms. After all, $\log \log \cdots \log x$ is eventually a number that's less than $1$, and the next logarithm gives a negative result. After that, you get a complex result. Continuing in this manner, the terms do not tend to zero.