Re: Ikedaian Cabalism

From: george murphy (gmurphy@raex.com)
Date: Thu Jun 28 2001 - 07:16:15 EDT

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    "Stein A. StrÝmme" wrote:

    > [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.

            Perhaps he said this with reference to the casual ways in which
    mathematicians of the 17th & 18th centuries handled series, in which case
    it's understandable. But divergent series can be very useful. In
    particular, there are _asymptotic_ series in x which for sufficiently
    large values of x give good approximations to a function and can be used
    for calculation of values of that function. The familiar Stirling
    approximation for x! is actually the first term of such a series.

    Shalom,

    George

    George L. Murphy
    http://web.raex.com/~gmurphy/
    "The Science-Theology Dialogue"



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