| 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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