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[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 Strømme <http://www.mi.uib.no/~stromme>

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