History of Grandi's Series - Euler

Euler

Further information: Euler on infinite series

Leonhard Euler treats 1 − 1 + 1 − 1 + · · · along with other divergent series in his De seriebus divergentibus, a 1746 paper that was read to the Academy in 1754 and published in 1760. He identifies the series as being first considered by Leibniz, and he reviews Leibniz's 1713 argument based on the series 1 − a + a2 − a3 + a4 − a5 + · · ·, calling it "fairly sound reasoning", and he also mentions the even/odd median argument. Euler writes that the usual objection to the use of 1/(1 + a) is that it does not equal 1 − a + a2 − a3 + a4 − a5 + · · · unless a is less than 1; otherwise all one can say is that

where the last remainder term does not vanish and cannot be disregarded as n is taken to infinity. Still writing in the third person, Euler mentions a possible rebuttal to the objection: essentially, since an infinite series has no last term, there is no place for the remainder and it should be neglected. After reviewing more badly divergent series like 1 + 2 + 4 + 8 + · · ·, where he judges his opponents to have firmer support, Euler seeks to define away the issue:

Yet however substantial this particular dispute seems to be, neither side can be convicted of any error by the other side, whenever the use of such series occurs in analysis, and this ought to be a strong argument that neither side is in error, but that all disagreement is solely verbal. For if in a calculation I arrive at this series 1 − 1 + 1 − 1 + 1 − 1 etc. and if in its place I substitute 1/2, no one will rightly impute to me an error, which however everyone would do had I put some other number in the place of this series. Whence no doubt can remain that in fact the series 1 − 1 + 1 − 1 + 1 − 1 + etc. and the fraction 1/2 are equivalent quantities and that it is always permitted to substitute one for the other without error. Thus the whole question is seen to reduce to this, whether we call the fraction 1/2 the correct sum of 1 − 1 + 1 − 1 + etc.; and it is strongly to be feared that those who insist on denying this and who at the same time do not dare to deny the equivalence have stumbled into a battle over words.

But I think all this wrangling can be easily ended if we should carefully attend to what follows…

Euler also used finite differences to attack 1 − 1 + 1 − 1 + · · ·. In modern terminology, he took the Euler transform of the sequence and found that it equalled 1⁄₂. As late as 1864, De Morgan claims that "this transformation has always appeared one of the strongest presumptions in favour of 1 − 1 + 1 − … being 1⁄₂."