"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.
George L. Murphy
"The Science-Theology Dialogue"
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