Transfinite Sums Related to a Sequence - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:10:04Z http://mathoverflow.net/feeds/question/46311 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46311/transfinite-sums-related-to-a-sequence Transfinite Sums Related to a Sequence Michael Burge 2010-11-17T01:06:15Z 2010-11-17T23:03:01Z <p>Hello,</p> <p>Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products of all $k+1$ tuples of elements taken from {$S_\beta| \beta &lt; \alpha$} if the sum exists. Also, define $(S+k)$ for $k \in \mathbb{N}$ to be the sequence <code>$(S+k)_n = S_{k+n}$</code>, essentially "dropping" the terms up to index $k$.</p> <p>For example, $S_\omega$ is the standard sum of a series from Calculus, $S_{\omega+1} = \sum_{n=1}(S_n \cdot \sum_{k=n+1}^\infty S_k)$, and in general $S_{\alpha+1} = \sum_{n=1}S_n (S+n)_\alpha$.</p> <p>I've managed to convince myself it has a few self-consistent properties:</p> <ul> <li>If $S_\alpha$ exists, then $(S+k)_\alpha$ exists.</li> <li>If $S_\alpha$ exists and $\beta &lt; \alpha$, then $S_\beta$ exists.</li> </ul> <p>My questions are:</p> <ol> <li><p>Has this concept been studied before, and if so can you provide references? I'm more concerned about the formal properties of these sequences, such as the algebra it generates. I suspect there's some paper or book on Generating Functions that would cover it. </p></li> <li><p>Given a set S of Complex numbers and an ordinal $\alpha$, you can create a new set $S_\alpha$ containing $s_\beta$ for every $\beta &lt;= \alpha$ and every sequence $s$ of numbers taken from $S$. Is there a paper that gives more detail on this operation? Interesting choices for $S$ might be the set of all roots of unity, or the set of zeros from some analytic function.</p></li> <li><p>Except for the constant 0 sequence, is it true that if you repeatedly transfinitely extend a sequence you will find a large enough ordinal where the sum does not exist? If so, what is the largest ordinal(or the supremum of all such ordinals) possible to extend a sequence to?</p></li> <li><p>My definition should work for countable ordinals, but I'm not entirely sure it's well-defined for anything larger. Is there a more natural/general definition I should be using that captures the concept better?</p></li> </ol> <p>Thank you,</p> <p>-- Michael Burge</p>