most recent 30 from http://mathoverflow.net 2013-05-21T19:52:07Z http://mathoverflow.net/feeds/user/21322 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88209/random-hyperharmonic-series tambay 2012-02-11T16:23:35Z 2012-02-11T18:00:42Z

<p>The Harmonic Series is defined as: $\sum_{n} \frac{1}{n}$ where $n=1,2,3,4....$. This series is known to be divergent.</p> <p>A generalization of this series can be made by raising each term to $p$: $\sum_{n} \frac{1}{n^p}$ which is also known as the hyperharmonic series and is known to be convergent when $p>1$.</p> <p>On the other hand, for $p=1$, if the signs of the terms are alternating the sum: $\sum_{n} \frac{(-1)^n}{n}$ is convergent and approaches $\ln{2}$.</p> <p>A natural extension would be to introduce randomness in the sign of each terms. $\sum_{n} \frac{\epsilon_{n}}{n}$ where $\epsilon_{n}$ is defined by the probability of its outcome: $P(\epsilon_{j} = 1)=P(\epsilon_{j} = -1)=1/2$. This is called the random harmonic series in Schmuland (http://www.stat.ualberta.ca/people/schmu/preprints/rhs.pdf).</p> <p>My question is: What would happen if we generalize this to the case of the random HYPERharmonic series? What would be the distribution of the result of the summation?</p>