Convergence of a general Bertrand series - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:45:43Z http://mathoverflow.net/feeds/question/10767 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10767/convergence-of-a-general-bertrand-series Convergence of a general Bertrand series Jérôme JEAN-CHARLES 2010-01-05T02:17:03Z 2010-01-05T02:59:24Z <p>Let $S= \sum 1/n log^1n log^2n log^3n ..log^{TL(n)}n$. </p> <p>Is it convergent when $n$ runs on integers say above 2 ? </p> <p>$log^i n$ denotes the i'th iterate of $log$ (in base 2 ) of $n$, $log^2n$ means $loglogn$ .</p> <p>$T(n)$ is the tower of $n$ (stack of $n$ 2's) that is $T(1)=2$ , $T(n+1)=2^{T(n)}$.</p> <p>$TL(n)$ is the <em>towerian log</em>:<br /> $TL(n) = Sup ( k : T(k) &lt;= n &lt; T(k+1) )$.</p> <p><strong>MOTIVATION</strong> : Generalizing the following that are called Bertrand series (I think):</p> <p>$\sum 1/n$ is the harmonic serie , $\sum 1/nlogn$ , $\sum 1/nlognlog^2n$ and $\sum 1/nlognlog^2nlog^3n$ are all known to be divergent.</p> <p>Here the product of iterated logs is pushed as far as possible and its size <strong>depends</strong> on the parameter $n$.</p> http://mathoverflow.net/questions/10767/convergence-of-a-general-bertrand-series/10769#10769 Answer by David Speyer for Convergence of a general Bertrand series David Speyer 2010-01-05T02:35:44Z 2010-01-05T02:35:44Z <p>The sum diverges. This is <a href="http://www.unl.edu/amc/a-activities/a7-problems/putnam/-pdf/2008s.pdf" rel="nofollow">Putnam Problem A4, 2008</a>. </p>