1 − 2 + 3 − 4 + ⋯

In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs.

The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, 3, ...), does not tend towards any finite limit.

Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation:

Starting in 1890, Ernesto Cesàro, Émile Borel and others investigated well-defined methods to assign generalized sums to divergent series—including new interpretations of Euler's attempts.

Divergence can also be shown directly from the definition: an infinite series converges if and only if the sequence of partial sums converges to a limit, in which case that limit is the value of the infinite series.

[2] Since the terms 1, −2, 3, −4, 5, −6, ... follow a simple pattern, the series 1 − 2 + 3 − 4 + ... can be manipulated by shifting and term-by-term addition to yield a numerical value.

If it can make sense to write s = 1 − 2 + 3 − 4 + ... for some ordinary number s, the following manipulations argue for s = 1⁄4:[3]

[4] In 1891, Ernesto Cesàro expressed hope that divergent series would be rigorously brought into calculus, pointing out, "One already writes (1 − 1 + 1 − 1 + ...)2 = 1 − 2 + 3 − 4 + ... and asserts that both the sides are equal to 1⁄4.

[1] The details on his summation method are below; the central idea is that 1 − 2 + 3 − 4 + ... is the Cauchy product (discrete convolution) of 1 − 1 + 1 − 1 + ... with 1 − 1 + 1 − 1 + ....

In the case where an = bn = (−1)n, the terms of the Cauchy product are given by the finite diagonal sums

With the result of the previous section, this implies an equivalence between summability of 1 − 1 + 1 − 1 + ... and 1 − 2 + 3 − 4 + ... with methods that are linear, stable, and respect the Cauchy product.

The series 1 − 1 + 1 − 1 + ... is Cesàro-summable in the weakest sense, called (C, 1)-summable, while 1 − 2 + 3 − 4 + ... requires a stronger form of Cesàro's theorem,[6] being (C, 2)-summable.

Since all forms of Cesàro's theorem are linear and stable,[7] the values of the sums are as calculated above.

There are two well-known generalizations of Cesàro summation: the conceptually simpler of these is the sequence of (H, n) methods for natural numbers n. The (H, 1) sum is Cesàro summation, and higher methods repeat the computation of means.

In the case of 1 − 2 + 3 − 4 + ..., his ideas are similar to what is now known as Abel summation: ... it is no more doubtful that the sum of this series 1 − 2 + 3 − 4 + 5 etc.

One can take the Taylor expansion of the right-hand side, or apply the formal long division process for polynomials.

To compute the Euler transform, one begins with the sequence of positive terms that makes up the alternating series—in this case 1, 2, 3, 4, ....

[14] Saichev and Woyczyński arrive at 1 − 2 + 3 − 4 + ... = 1⁄4 by applying only two physical principles: infinitesimal relaxation and separation of scales.

To be precise, these principles lead them to define a broad family of "φ-summation methods", all of which sum the series to 1⁄4: This result generalizes Abel summation, which is recovered by letting φ(x) = exp(−x).

The general statement can be proved by pairing up the terms in the series over m and converting the expression into a Riemann integral.

The threefold Cauchy product of 1 − 1 + 1 − 1 + ... is 1 − 3 + 6 − 10 + ..., the alternating series of triangular numbers; its Abel and Euler sum is 1⁄8.

[16] The fourfold Cauchy product of 1 − 1 + 1 − 1 + ... is 1 − 4 + 10 − 20 + ..., the alternating series of tetrahedral numbers, whose Abel sum is 1⁄16.

This sum became an object of particular ridicule by Niels Henrik Abel in 1826: Divergent series are on the whole devil's work, and it is a shame that one dares to found any proof on them.

[18]Cesàro's teacher, Eugène Charles Catalan, also disparaged divergent series.

Under Catalan's influence, Cesàro initially referred to the "conventional formulas" for 1 − 2n + 3n − 4n + ... as "absurd equalities", and in 1883 Cesàro expressed a typical view of the time that the formulas were false but still somehow formally useful.

Finally, in his 1890 Sur la multiplication des séries, Cesàro took a modern approach starting from definitions.

[19] The series are also studied for non-integer values of n; these make up the Dirichlet eta function.

Euler had already become famous for finding the values of these functions at positive even integers (including the Basel problem), and he was attempting to find the values at the positive odd integers (including Apéry's constant) as well, a problem that remains elusive today.

The eta function in particular is easier to deal with by Euler's methods because its Dirichlet series is Abel summable everywhere; the zeta function's Dirichlet series is much harder to sum where it diverges.

[20] For example, the counterpart of 1 − 2 + 3 − 4 + ... in the zeta function is the non-alternating series 1 + 2 + 3 + 4 + ..., which has deep applications in modern physics but requires much stronger methods to sum.