http://www.openproblemgarden.org/op/pushing_bertrand_series Comments for "pushing Bertrand series" en http://www.openproblemgarden.org/op/pushing_bertrand_series <table cellspacing="10"> <tr> <td> Author(s): <a href="/kpz_equation_central_limit_theorem"></a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/analysis">Analysis</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Question</b>&nbsp;&nbsp; <b>Definitions</b>: For <img class="teximage" src="/files/tex/cf0b36d79ceee4731552ce40cf715371dc83341e.png" alt="$ n \gt 1 $" /> : Let <img class="teximage" src="/files/tex/cc8ce1649c9fb0802349c8683ad9ea5b90b304f1.png" alt="$ K(n) $" /> be the last <img class="teximage" src="/files/tex/ca49c241ece07915c97a31774a977841c6f0414c.png" alt="$ i $" /> such that the i'th iteration of log composition on <img class="teximage" src="/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png" alt="$ n $" /> is still above 1.</p> <p>Remark: K(n) is the "towerian log" where <img class="teximage" src="/files/tex/4220b0da3c2bdb4b121f04b7baf72a9a0624cb4c.png" alt="$ tower(n) = 2^{tower(n-1)} $" /> and <img class="teximage" src="/files/tex/105dc82b52c9f3b79c4c11aedd215a6dc2971add.png" alt="$ tower(0)=1 $" />.</p> <p>Let <img class="teximage" src="/files/tex/4232f17dd19af4228a83c653a3a277cd42dc887d.png" alt="$ p(n) = n.logn.loglogn......log(log(log... (n))) $" /> where the last log is iterated <img class="teximage" src="/files/tex/cc8ce1649c9fb0802349c8683ad9ea5b90b304f1.png" alt="$ K(n) $" /> times.</p> <p><b>PROBLEM :</b> Is <img class="teximage" src="/files/tex/933657e52a68540a310b24560a42cd74015ee63b.png" alt="$ \sum_{i=1}^{n} \frac{1}{p(n)} $" />. convergent or divergent.</p> <p><b>REMARKS:</b> The log could be in base 2 this clearly does not affect the result.</p> <p>The log could be replaced by ceil(log) or floor(log) , this may affect the result.</p> <p><b>MOTIVATION: </b> the sum of <img class="teximage" src="/files/tex/b641b46ffef39a309ca18f4c26d7ad79be3919ad.png" alt="$ 1/n $" /> ( harmonic serie is divergent) , the same is true with <img class="teximage" src="/files/tex/f6ae4b83c8538afcf7e0d50c615cb97d0c225454.png" alt="$ \frac{1}{nlogn} $" /> and <img class="teximage" src="/files/tex/8ee36d9b2982b3538313b5aec7fc5ad70f079995.png" alt="$ \frac{1}{nlognloglogn} $" /> etc (called bertrand series I think) . </p> <p><b>HISTORY:</b> I made it up, it should have been thought of somewhere else (my culture in analysis is small).</p> </div> </tr></td> </table> </td> </tr> </table> Analysis http://www.openproblemgarden.org/op/pushing_bertrand_series#comment Wed, 20 May 2009 01:54:42 +0200 Jerome JEAN-CHARLES 36943 at http://www.openproblemgarden.org