**Previous message:**Stein A. Strømme: "Re: Ikedaian Cabalism"**In reply to:**Stein A. Strømme: "Re: Ikedaian Cabalism"**Next in thread:**Dick Fischer: "Re: Ikedaian Cabalism"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

"Stein A. Strømme" wrote:

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