Summability theory

From: Nick Singer (nsinger2_at_cox.net)
Date: 09/21/04

  • Next message: Robert Israel: "Re: Summability theory" 
    Date: 21 Sep 2004 16:45:02 -0400

    Consider the series sum_{n>=1} f(n)/n, where f maps the positive
    integers into the two-element set {-1,1}.

    (1) What are necessary and sufficient conditions on f to ensure
    convergence? Partial "necessary" answer from Cesaro: the limit of
    the ratio of positive terms to negative terms, if it exists, must be
    1. Possibly Littlewood's theorem could help here: it's summable
    if the terms are O(1/n) [yes, they are], and the series is Abel
    summable [maybe...]. How about something of the form
    sum {n=1 to N} f(n) = O(log N), or some such slow-growing function of
    N ?

    Now consider a generalization: the series sum_{n>=1} f(n)/g(n), where
    f is as before and g is any positive, strictly increasing function,
    e.g. g(n) = log(log(n+20)).

    (2) Now what are necessary and sufficient conditions on f to ensure
    convergence? The Cesaro condition is still necessary. Littlewood's
    theorem doesn't help anymore if g(n) grows more slowly than n.

    Thanks for your help,
    Nick

  • Next message: Robert Israel: "Re: Summability theory"

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