Re: Ikedaian Cabalism

From: Stein A. StrÝmme (stromme@mi.uib.no)
Date: Thu Jun 28 2001 - 02:50:10 EDT

  • Next message: george murphy: "Re: Ikedaian Cabalism"

    [george murphy]

    | Yes, the apparently innocuous procedure of group terms in an
    | infinite series in the ways required by the above "proof" is not
    | valid unless the series in question have the required sorts of
    | convergence properties, which of course the manifestly divergent
    | (because its sequence of partial sums alternate between 1 & 0 & thus
    | have no limit) series 1 - 1 + 1 - .... doesn't have.
    | Before modern ideas about convergence were well developed,
    | however, quite competent mathematicians handled divergent series in
    | ways that would earn a calculus student today an F, & even today
    | there are consistent ways of _defining_ sums for divergent series.
    | E.g., if the sum is defined as the limit of the _mean_ of the
    | sequence of partial sums (Cesaro summation) then the above series
    | has the value 1/2, which is also the value of the function 1/(1 +
    | x), which equals 1 - x + x^2 - x^3 + .... when x < 1. When x = 1
    | then 1/(1+x) = 1/2 and 1 - x + x^2 - x^3 + .... = 1 - 1 + 1 - 1 +
    | .... .

    Abel (the Norwegian mathematician, not Cain's brother) wrote that
    divergent series is the work of the devil.

    -- 
    Stein Arild StrÝmme  <http://www.mi.uib.no/~stromme>
    



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