From: "David Petry" Subject: Re: "Largest" integrals and "slowest" series. Date: Wed, 10 Jan 2001 04:29:16 -0800 Newsgroups: sci.math Summary: Very slowly converging (or diverging) series Bill Taylor wrote >Is it possible to doctor up the example to give a similar-looking series >that CON-verges more slowly than any series in the OTHER half of the table? Given a sequence { a_k } such that sum(a_k) diverges, let S_n = sum(k=1..n)( a_k ). Then sum( a_k / (S_k*S_{k-1} ) ) converges. (For kicks, let S_0 = 1) Furthermore, if the first sum diverges very very slowly, the second sum converges very very slowly. ============================================================================== From: renfrod@central.edu (Dave L. Renfro) Subject: Re: "Largest" integrals and "slowest" series. Date: 10 Jan 2001 15:18:16 -0500 Newsgroups: sci.math Bill Taylor [sci.math 10 Jan 2001 04:34:00 GMT] wrote > I don't know how I missed seeing this post the first time!! > Thanks for reposting it Dave, that's *just* the sort of thing I > was after when I referred to "some sort of omega-ly defined > series". Brilliant. > > I'm sure that's the sort of thing I tried to find myself, too, > but couldn't handle the -vergence calculations decently. I'll > certainly check out your refs. Note to others reading this: I didn't come up with the example Bill is talking about. The example I gave was published by R. P. Agnew in a 1947 Amer. Math. Monthly paper (details in my earlier post). > > One last question remains, though. > > Is it possible to doctor up the example to give a similar-looking > series that CON-verges more slowly than any series in the OTHER > half of the table? > > That would be ultra-kool!! > > Hmmm... maybe if we always raised to (1+eps) power just the > *last* term in the product of finite-but-variable length terms? sol!ikastan's (= Ilias Kastanas) Sept. 11 reply to my Sept. 10 sci.math post at contains (among other remarks): ****************************************************************** I haven't seen Ash's paper, but an easy proof is: let s_n = c_1 + ... + c_n; if Sum c_n diverges, then Sum c_n/s_n also diverges. Let t_n = a_n + a_n+1 + ...; if Sum a_n converges, then Sum a_n/sqrt(t_n) also converges. ****************************************************************** Right now I'm about 900 miles away from all my books and papers, and nowhere near a college library, so I can't compare what Ilias Kastanas wrote with what is in Ash's paper. > > Anyway - thanks again. > > My only remaining grouch is that our library's copy of Hardy's > booklet on the DuBois stuff has been vanished for some time. > I used to love perusing it! I mentioned (in my other post) a neat historical survey paper by Gordon Fisher on Paul du Bois-Reymond's work with series (Bois-Reymond was the person who discovered that there is no "countably obtainable" boundary between convergent and divergent series of nonnegative terms). [To answer another question in this thread, I think the completion of a Hausdorff $\eta_{\alpha}$-ordered set (see Gillman and Jerison's "Rings of Continuous Functions") for an appropriate $\alpha$ will give you the boundary itself, but not very constructively.] Regarding these Bois-Reymond papers, at least one of them is available on-line. Go to the Zbl web page , enter Bois-Reymond into the "Author" space. Then, near the bottom of this web page, type 100 into the "Display ... entries per page" space, and finally select complete just to the right of this. [The options are "short" and "complete"; "complete" will give you the available on-line JFM reviews.] After doing this, select "Start Retrieval". [If you have a slow phone-line connection (what I'm using), you might want to alter these instructions by selecting "short" rather than "complete", and then get the information (the JFM review and the icon link to a digitally scanned version of the paper, when one is available) by clicking on the JFM citation number.] You'll get a lot of information about Bois-Reymond's publications. Most of these papers have JFM reviews (in German) on-line, and quite a few of these papers have their full text digitally scanned and on-line (see the "Link to full text" icon that appears at the end of some of the JFM reviews). By the way, if you can't read German (I can't), you can try this (what I do) --->>>> Select and copy the JFM review text you want to translate, go to , paste the copied text into the appropriate window, select "German to English", and then click "Translate". It's crude (and its really awful with math terms), but if you have even the slightest clue as to what the paper is about and you can't read German at all, you can usually learn a good deal more doing this. Here are two of Bois-Reymond's papers that deal with convergence/divergence boundary matters: Paul du Bois-Reymond, "Eine neue Theorie der Convergenz und Divergenz von Reihen mit positiven Gliedern" ["A new theory of convergence and divergence of series with positive terms"], Borchard J. 76 (1873), 61-91. [The JFM 05.0128.01 review is available on-line, but a digitally scanned copy of the paper isn't provided.] Paul du Bois-Reymond, "Ueber die Paradoxen des Infinitär-Calcüls" ["On the paradoxes of the infinitary calculus"], Math. Annalen 11 (1877), 150-167. [Both the JFM 09.0297.01 review and a a digitally scanned copy of the paper are available.] To digress a moment, Bois-Reymond was the person who gave the first published account of the Weierstrass nowhere differentiable continuous function. The JFM 06.0241.01 review of this paper is available on-line, but a digitally scanned copy of the paper isn't provided. However, this function and (I believe) essentially the same proof that Weierstrass gave (as given in Bois-Reymond's paper) can be found in English on pages 58-60 of James Harkness, "A Treatise on the Theory of Functions", 1893. A digitally scanned copy of Harkness's book can be found at the next URL that I give. Regarding G. H. Hardy's book "Orders of Infinity", I thought it would be among the digitally scanned books at , but it isn't there (as of 1:00 P.M. CST Jan. 10, 2001). However, E. Borel has written a lot of stuff about growth rates of functions and sequences (I learned this from researching the "Borel rarefaction" classification of measure zero sets), and there are two digitally scanned books by Borel (both in French) that deal with series convergence/divergence matters. Emile Borel, "Lecons sur les Series a Termes Positifs" ["Lectures on Series with Positive Terms"], 1902. [The table of contents is on page vii.] Emile Borel, "Lecons Sur la Theorie de la Croissance" [Lectures on the Theory of Growth"], 1910. [The table of contents is on page vii.] Here's another digitally scanned book that might be of interest: Paul du Bois-Reymond, "Die Allgemeine Functionentheorie" (I), 1882. [The table of contents is on page xi.] Dave L. Renfro ============================================================================== From: "Rainer Rosenthal" Subject: Re: "Largest" integrals and "slowest" series. Date: Wed, 10 Jan 2001 21:45:13 +0100 Newsgroups: sci.math Dave L. Renfro wrote in message news:sv8l6zlrgaut@forum.mathforum.com... | Rainer Rosenthal | [sci.mathMon, 8 Jan 2001 22:02:30 +0100] | | | and, among the various results, I found the person I was thinking | of (Eric K. van Dowen). A quick search at the Zbl web page (I don't | have access to Math. Reviews on-line, aside from | ) at | | | | gave me the specific paper and an on-line review for it: | | Eric K. van Dowen, "Finitely additive measures on N", Topology | Appl. 47 (1992), 223-268. MR 94c:28004; Zbl 762.28010 | | I managed to also find another paper doing some key word searchs | at the Zbl web page, but I don't know if it has things of interest | to you (the on-line Zbl review is in French): | | M. Deza and Paul Erdos, "Extension de quelques theoremes sur les | densites de series d'elements de $N$ a des series de sous-ensembles | finis de $N$", J. Discrete Math. 12 (1975), 295-308. Zbl 308.05004 | Hello Dave, first of all I am very pleased to be taken serious with my fancies ! Next I thank you very much for your friendly answer and these many references. There was only a reference to "Alan H. Mekler, Finitely additive measures on N and the additive property". My French is smaller than petit and may be someone from the audience will think: "Well, Rainer did not talk complete nonsense and Dave L. Renfro gave these fine references - so let me see if I do not have another one ...". Thanks in advance ( I will try my French ...) Rainer