Question

Let $ S= \sum 1/n log^1n log^2n log^3n ..log^{TL(n)}n $.

Is it convergent when $n$ runs on integers say above 2 ?

$log^i n$ denotes the i'th iterate of $log$ (in base 2 ) of $n$, $log^2n$ means $loglogn$ .

$T(n)$ is the tower of $n$ (stack of $n$ 2's) that is $T(1)=2$ , $T(n+1)=2^{T(n)}$.

$TL(n)$ is the towerian log:
$ TL(n) = Sup ( k : T(k) <= n < T(k+1) ) $.

MOTIVATION : Generalizing the following that are called Bertrand series (I think):

$\sum 1/n$ is the harmonic serie , $\sum 1/nlogn$ , $\sum 1/nlognlog^2n $ and $\sum 1/nlognlog^2nlog^3n $ are all known to be divergent.

Here the product of iterated logs is pushed as far as possible and its size depends on the parameter $n$.

Answer 1

The sum diverges. This is Putnam Problem A4, 2008.