Series with iterated log's: does it converge?

Series with iterated log’s: does it converge?

This came up in our office today. Let

$$f(x) = \begin{cases} x & \mbox{if } x\leq 1 \cr x\cdot f(\ln(x)) & \mbox{otherwise}\end{cases}$$

Does this series converge?

$$ \sum_{n=1}^\infty \frac{1}{f(n)} $$

flag	asked Nov 2 at 3:06 Will Gunther 1

This problem came up on a recent Putnam. The answer was that it diverges at a similar rate to the function $log_*$. – zeb Nov 2 at 3:12

2008 A4, except it had $f(x)=x$ for $x\le e$. Solution available at amc.maa.org/a-activities/a7-problems/putnam/-pdf/… – Gerry Myerson Nov 2 at 5:01