Leonhard Euler reasoned that 1/4= 1 -2 + 3 -4 . . . in the 1700s and Niels Henrik Abel found a more rigorous method to confirm Euler's result, as did others. This odd result (in which a sum of a divergent series is assigned a total different from any of its partial sums that diverge to large positive and negative numbers alternately), is even more odd because one can apply Abel's method of summing divergent series to a more general series 1 − 2n + 3n − 4n + etc., which reduces to the divergent infinite series 1 - 2 + 3 - 4 . . . which Euler and Abel had found to be equal to 1/4 in case of n=1, and also to the divergent infinite series called Grandi's series, 1 -1 +1 -1 . . . which various mathematicians had found to be equal to 1/2 in the case of n=0. Yet, the solution of the more general series 1 − 2n + 3n − 4n + etc., evaluated by Abel's method, always had the more intuitive value of zero, regardless of the value for n used, contradicting another sum of the same divergent series calculated using the same method!
Of course, no partial sum of any of these series ever has a non-integer or zero value, so one of a half a dozen methods of summing up an infinite series can be used. One can compute a sum by rearranging terms of the sums or over several sets of the sums in clever ways so that most of term terms cancel out. Or, one can compute a Cesàro sum (named after Ernesto Cesaro) by computing the arithmetic means of the partial sums of the series. Or, one can substitute different subsets of the infinite series for terms of convergent infinite series that have well defined values and add up the values of the convergent series that can be combined to create a divergent series, or take the limit of a convergent infinite series sum formula beyond the values for which it is well defined, as Abel's did.
Abel's method was to use the sum of the infinite series below, which is convergent for x with an absolute value of less than 1, and to take the limit of that result for the case of x=1 where the series becomes divergent instead.
Another conceptual approach to summing the value of a divergent infinite series is invert the process of adding up partial sums and taking them to a limit as the number of terms approach infinity. Rather than looking at the limit of the partial sums as the number of terms approaches infinity, one can look at the trendline that the partial sums approach in the limit from infinity to zero (in practice, starting with a vary large N partial sum and tracking the trend of the partial sum downward), with the value of the sum of the series in the limit as it has zero terms defined as the sum of the terms in the series.
Abel assigned the title of the Devil's Arithmetic to the entire business of this different kind of arithmetic of the assigning of values to the sum of the terms of divergent infinite series, more fittingly than Yolen's appropriation of the term, in a moment in 1826 recounted in the translation of the original French below:
Divergent series are on the whole devil's work, and it is a shame that one dares to found any proof on them. One can get out of them what one wants if one uses them, and it is they which have made so much unhappiness and so many paradoxes. Can one think of anything more appalling than to say thatAccording to Grattan-Guinness, Ivor, The development of the foundations of mathematical analysis from Euler to Riemann (1970) at 80, and corroborated in a slightly different translation from the French with the same sense by Markushevich, A.I. Series: fundamental concepts with historical exposition (3rd revised edition (1961) in Russian translated into English (1967) at 48.
where n is a positive number. Here's something to laugh at, friends.
- 0 = 1 − 2n + 3n − 4n + etc.
None of these mathematicians and none of the contradictory results were "wrong" in a crude sense. The crude answer is that divergent infinite series don't have well defined summation values unless you adopt some sort of non-obvious definition of their summed values, and that no definition of this inherently counterintuitive puzzle can be wholly satisfactory.
Each algorithm for defining the sum leads to a particular result, although not necessarily a unique result, as Abel discovered the hard way (a more familiar function which has a well defined result that is often not unique, is the square root of a number, which usually had two well defined answers).
This would all be just obscure mathematical game playing were it not for the fact that at the very heart of quantum mechanics is the need to sum up the aggregate probability of infinite number of paths that a particle could take to get from point A to point B in order to know the probability that it will actually go from point A to point B, which is well defined for each of those particular paths. Some possible questions that come up using these formulas or beyond the Standard Model variants of them, sometimes generate divergent infinite series that have to be summed up. So, figuring out a definition that produces the right answer out of the various possible definitions you could use, can matter. And, when a definition doesn't produce unique answers, this makes reality itself confounding.
Footnote on Abel's death and TB:
Less than three years after his devil's arithmetic statement, in 1829, Abel died at the age of 47 of tuberculosis (TB). He spent his last Christmas in 1828, with his fiancée, while suffering seriously from this illness, but never ended up marrying her. He also didn't live to receive a letter that came two days after his death informing the then unemployed man that he had received an appointment as a professor in Berlin. Practical applications for his work, which are important in modern fundamental physics, came almost entirely after his death in the 20th century.
The discovery that TB was transmitted by a germ allowed for primitive quarantine measures that reduced its incidence by half in the late 1800s, and by 1921, a TB vaccine had been invented, but was not widely used. An effective antibiotic to treat the disease was not invented until 1944, but only in 1952 with the development of an oral antibiotic was effective treatment for TB available on a widespread basis.