# 1 − 2 + 3 − 4 + ⋯ Information

*https://en.wikipedia.org/wiki/1_−_2_+_3_−_4_+_⋯*

In
mathematics, **1 − 2 + 3 − 4 + ···** is an
infinite series whose terms are the successive
positive integers, given
alternating signs. Using
sigma summation notation the sum of the first *m* terms of the series can be expressed as

The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, ...), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation:

A rigorous explanation of this equation would not arrive until much later. 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. Many of these summability methods easily assign to 1 − 2 + 3 − 4 + ... a "value" of 1/4. Cesàro summation is one of the few methods that do not sum 1 − 2 + 3 − 4 + ..., so the series is an example where a slightly stronger method, such as Abel summation, is required.

The series 1 − 2 + 3 − 4 + ... is closely related to
Grandi's series 1 − 1 + 1 − 1 + .... Euler treated these two as special cases of the more general sequence 1 − 2^{n} + 3^{n} − 4^{n} + ..., where *n* = 1 and *n* = 0 respectively. This line of research extended his work on the
Basel problem and leading towards the
functional equations of what are now known as the
Dirichlet eta function and the
Riemann zeta function.

## Divergence

The series' terms (1, −2, 3, −4, ...) do not approach
0; therefore 1 − 2 + 3 − 4 + ... diverges by the
term test. Divergence can also be shown directly from the definition: an infinite series converges if and only if the sequence of partial sums
converges to limit, in which case that limit is the value of the infinite series. The partial sums of 1 − 2 + 3 − 4 + ... are:^{
[1]}

**1**,

1 − 2 = **−1**,

1 − 2 + 3 = **2**,

1 − 2 + 3 − 4 = **−2**,

1 − 2 + 3 − 4 + 5 = **3**,

1 − 2 + 3 − 4 + 5 − 6 = **−3**,

The sequence of partial sums shows that the series does not converge to a particular number: for any proposed limit *x*, there exists a point beyond which the subsequent partial sums are all outside the interval [*x*−1, *x*+1]), so 1 − 2 + 3 − 4 + ... diverges.

The partial sums include every integer exactly once—even 0 if one counts the
empty partial sum—and thereby establishes the
countability of the set of
integers.^{
[2]}

## Heuristics for summation

### Stability and linearity

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]}

So .

Although 1 − 2 + 3 − 4 + ... does not have a sum in the usual sense, the equation *s* = 1 − 2 + 3 − 4 + ... = 1⁄4 can be supported as the most natural answer if such a sum is to be defined. A
generalized definition of the "sum" of a divergent series is called a
summation method or
summability method. There are many different methods and it is desirable that they share some
properties of ordinary summation. What the above manipulations actually prove is the following: Given any summability method that is
linear and stable and sums the series 1 − 2 + 3 − 4 + ..., the sum it produces is 1⁄4.^{
[4]} Furthermore, since

^{ [4]}

### Cauchy product

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."^{
[5]} For Cesàro, this equation was an application of a theorem he had published the previous year, which is the first theorem in the history of summable divergent series.^{
[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 + ....

The Cauchy product of two infinite series is defined even when both of them are divergent. In the case where *a*_{n} = *b*_{n} = (−1)^{n}, the terms of the Cauchy product are given by the finite diagonal sums

The product series is then

Thus a summation method that respects the Cauchy product of two series — and assigns to the series 1 − 1 + 1 − 1 + ... the sum 1/2 — will also assign to the series 1 − 2 + 3 − 4 + ... the sum 1/4. 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.

Cesàro's theorem is a subtle example. 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.

## Specific methods

### Cesàro and Hölder

To find the (C, 1) Cesàro sum of 1 − 2 + 3 − 4 + ..., if it exists, one needs to compute the arithmetic means of the partial sums of the series. The partial sums are:

and the arithmetic means of these partial sums are:

This sequence of means does not converge, so 1 − 2 + 3 − 4 + ... is not Cesàro summable.

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. Above, the even means converge to 1⁄2, while the odd means are all equal to 0, so the means *of the means* converge to the average of 0 and 1⁄2, namely 1⁄4.^{
[8]} So 1 − 2 + 3 − 4 + ... is (H, 2) summable to 1⁄4.

The "H" stands for
Otto Hölder, who first proved in 1882 what mathematicians now think of as the connection between
Abel summation and (H, *n*) summation; 1 − 2 + 3 − 4 + ... was his first example.^{
[9]} The fact that 1⁄4 is the (H, 2) sum of 1 − 2 + 3 − 4 + ... guarantees that it is the Abel sum as well; this will also be proved directly below.

The other commonly formulated generalization of Cesàro summation is the sequence of (C, *n*) methods. It has been proven that (C, *n*) summation and (H, *n*) summation always give the same results, but they have different historical backgrounds. In 1887, Cesàro came close to stating the definition of (C, *n*) summation, but he gave only a few examples. In particular, he summed 1 − 2 + 3 − 4 + ..., to 1⁄4 by a method that may be rephrased as (C, *n*) but was not justified as such at the time. He formally defined the (C, n) methods in 1890 in order to state his theorem that the Cauchy product of a (C, *n*)-summable series and a (C, *m*)-summable series is (C, *m* + *n* + 1)-summable.^{
[10]}

### Abel summation

In a 1749 report, Leonhard Euler admits that the series diverges but prepares to sum it anyway:

... when it is said that the sum of this series 1 − 2 + 3 − 4 + 5 − 6 etc. is 1⁄4, that must appear paradoxical. For by adding 100 terms of this series, we get −50, however, the sum of 101 terms gives +51, which is quite different from 1⁄4 and becomes still greater when one increases the number of terms. But I have already noticed at a previous time, that it is necessary to give to the word

suma more extended meaning ...^{ [11]}

Euler proposed a generalization of the word "sum" several times. 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. is 1⁄4; since it arises from the expansion of the formula 1⁄(1+1)

^{2}, whose value is incontestably 1⁄4. The idea becomes clearer by considering the general series 1 − 2x+ 3x^{2}− 4x^{3}+ 5x^{4}− 6x^{5}+&c.that arises while expanding the expression 1⁄(1+x)^{2}, which this series is indeed equal to after we setx= 1.^{ [12]}

There are many ways to see that, at least for
absolute values |*x*| < 1, Euler is right in that

*x*) twice or squaring the geometric series 1 −

*x*+

*x*

^{2}− .... Euler also seems to suggest differentiating the latter series term by term.

^{ [13]}

In the modern view, the
generating function 1 − 2*x* + 3*x*^{2} − 4*x*^{3} + ... does not define a
function at *x* = 1, so that value cannot simply be substituted into the resulting expression. Since the function is defined for all |*x*| < 1, one can still take the limit as *x* approaches 1, and this is the definition of the Abel sum:

### Euler and Borel

Euler applied another technique to the series: the
Euler transform, one of his own inventions. 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, .... The first element of this sequence is labeled *a*_{0}.

Next one needs the sequence of
forward differences among 1, 2, 3, 4, ...; this is just 1, 1, 1, 1, .... The first element of *this* sequence is labeled Δ*a*_{0}. The Euler transform also depends on differences of differences, and higher
iterations, but all the forward differences among 1, 1, 1, 1, ... are 0. The Euler transform of 1 − 2 + 3 − 4 + ... is then defined as

In modern terminology, one says that 1 − 2 + 3 − 4 + ... is Euler summable to 1⁄4.

The Euler summability also implies
Borel summability, with the same summation value, as it does in general.^{
[14]}

### Separation of scales

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:

- If
*φ*(*x*) is a function whose first and second derivatives are continuous and integrable over (0, ∞), such that φ(0) = 1 and the limits of φ(*x*) and*x*φ(*x*) at +∞ are both 0, then^{ [15]}

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. For the latter step, the
corresponding proof for 1 − 1 + 1 − 1 + ... applies the
mean value theorem, but here one needs the stronger Lagrange form of
Taylor's theorem.

## Generalization

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.

Another generalization of 1 − 2 + 3 − 4 + ... in a slightly different direction is the series 1 − 2^{n} + 3^{n} − 4^{n} + ... for other values of *n*. For positive integers *n*, these series have the following Abel sums:^{
[17]}

*B*

_{n}are the Bernoulli numbers. For even

*n*, this reduces to

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 that

0 = 1 − 2^{2n}+ 3^{2n}− 4^{2n}+ etc.where

nis a positive number. Here's something to laugh at, friends.^{ [18]}

Cesàro's teacher,
Eugène Charles Catalan, also disparaged divergent series. Under Catalan's influence, Cesàro initially referred to the "conventional formulas" for 1 − 2^{n} + 3^{n} − 4^{n} + ... 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. Part of Euler's motivation for studying series related to 1 − 2 + 3 − 4 + ... was the
functional equation of the eta function, which leads directly to the functional equation of the
Riemann zeta 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.

## See also

## References

- ^
^{a}^{b}Hardy, p. 8 **^**Beals, p. 23**^**Hardy (p. 6) presents this derivation in conjunction with evaluation of Grandi's series 1 − 1 + 1 − 1 + ....- ^
^{a}^{b}Hardy, p. 6 **^**Ferraro, p. 130.**^**Hardy, p. 3; Weidlich, pp. 52–55.**^**Alabdulmohsin 2018.**^**Hardy, p. 9. For the full details of the calculation, see Weidlich, pp. 17–18.**^**Ferraro, p. 118; Tucciarone, p. 10. Ferraro criticizes Tucciarone's explanation (p. 7) of how Hölder himself thought of the general result, but the two authors' explanations of Hölder's treatment of 1 − 2 + 3 − 4 + ... are similar.**^**Ferraro, pp. 123–128.**^**Euler et al., p. 2. Although the paper was written in 1749, it was not published until 1768.**^**Euler et al., pp. 3, 25.**^**For example, Lavine (p. 23) advocates long division but does not carry it out; Vretblad (p. 231) calculates the Cauchy product. Euler's advice is vague; see Euler et al., pp. 3, 26. John Baez even suggests a category-theoretic method involving multiply pointed sets and the quantum harmonic oscillator. Baez, John C. Euler's Proof That 1 + 2 + 3 + ... = −1/12 (PDF). Archived 2017-10-13 at the Wayback Machine math.ucr.edu (December 19, 2003). Retrieved on March 11, 2007.**^**Shawyer and Watson, p. 32**^**Saichev and Woyczyński, pp. 260–264.**^**Kline, p. 313.**^**Hardy, p. 3; Knopp, p. 491**^**Grattan-Guinness, p. 80. See Markushevich, p. 48, for a different translation from the original French; the tone remains the same.**^**Ferraro, pp. 120–128.**^**Euler et al., pp. 20–25.

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