1 − 2 + 3 − 4 + · · ·

In mathematics, 1 − 2 + 3 − 4 + … is the 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

: $sum_{n=1}^m n(-1)^{n-1}.$

The infinite series diverges, meaning that its sequence of partial sums, nowrap|(1, −1, 2, −2, …), does not tend towards any finite limit. Equivalently, one says that nowrap|1 − 2 + 3 − 4 + … lacks a sum.

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

: $1-2+3-4+cdots=frac{1}{4}.$

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 nowrap|1 − 2 + 3 − 4 + … a "sum" of frac|1|4 after all. Cesàro summation is one of the few methods that does not sum nowrap|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 nowrap|1 − 1 + 1 − 1 + …. Euler treated these two as special cases of nowrap|1 − 2^"n" + 3^"n" − 4^"n" + … for arbitrary "n", a line of research extending his work on the Basel problem and leading towards the functional equations of what we now know as the Dirichlet eta function and the Riemann zeta function.

Divergence

The series' terms (1, −2, 3, −4, …) do not approach 0; therefore nowrap|1 − 2 + 3 − 4 + … diverges by the term test. For later reference, it will also be useful to see the divergence on a fundamental level. By definition, the convergence or divergence of an infinite series is determined by the convergence or divergence of its sequence of partial sums, and the partial sums of nowrap|1 − 2 + 3 − 4 + … are: [Hardy p.8] :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,:…This sequence is notable for visiting every integer once—even 0 if one counts the empty partial sum—and thereby establishing the countability of the set $mathbb{Z}$ of integers. [Beals p.23] It clearly does not settle down and converge to a particular number, so nowrap|1 − 2 + 3 − 4 + … diverges.

Heuristics for summation

tability and linearity

Since the terms 1, −2, 3, −4, 5, −6, … follow a simple pattern, the series nowrap|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 nowrap|1="s" = 1 − 2 + 3 − 4 + … for some ordinary number "s", the following manipulations argue for nowrap|1="s" = frac|1|4: [Hardy (p.6) presents this derivation in conjunction with evaluation of Grandi's series nowrap|1 − 1 + 1 − 1 + ….]

$egin{array}{rclllll}4s&=& &(1-2+3-4+cdots) & +(1-2+3-4+cdots) & +(1-2+3-4+cdots) &+(1-2+3-4+cdots) \ &=& &(1-2+3-4+cdots) & +1+(-2+3-4+5+cdots) & +1+(-2+3-4+5+cdots) &-1+(3-4+5-6cdots) \ &=&1+ [&(1-2-2+3) & +(-2+3+3-4) & +(3-4-4+5) &+(-4+5+5-6)+cdots] \ &=&1+ [&0+0+0+0+cdots] \4s&=&1end{array}$ So $s=frac{1}{4}$ . This derivation is depicted graphically on the right.

Although 1 − 2 + 3 − 4 + … does not have a sum in the usual sense, the equation nowrap|1="s" = 1 − 2 + 3 − 4 + … = frac|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, which sums some subset of all possible series. There are many different methods (some of which are described below) that are characterized by the properties that they share with ordinary summation. What the above manipulations actually prove is the following: Given any summability method that is linear and stable and sums the series nowrap|1 − 2 + 3 − 4 + …, the sum it produces is frac|1|4. Furthermore, since

$egin{array}{rcllll}2s & = & &(1-2+3-4+cdots) & + & (1-2+3-4+cdots) \ & = & 1 + &(-2+3-4+cdots) & + 1 - 2 & + (3-4+5cdots) \ & = & 0 + &(-2+3)+(3-4)+ (-4+5)+cdots \frac{1}{2}& = & &1-1+1-1cdots \end{array}$

such a method must also sum Grandi's series as nowrap|1=1 − 1 + 1 − 1 + … = frac|1|2.

Cauchy product

In 1891, Ernesto Cesàro expressed hope that divergent series would be rigorously brought into calculus, pointing out, "One already writes nowrap|1=(1 − 1 + 1 − 1 + …)² = 1 − 2 + 3 − 4 + … and asserts that both the sides are equal to frac|1|4." [Ferraro, p.130.] For Cesàro, this equation was an application of a theorem he had published the previous year, one that may be identified as the first theorem in the history of summable divergent series. The details on his summation method are below; the central idea is that nowrap|1 − 2 + 3 − 4 + … is the Cauchy product of nowrap|1 − 1 + 1 − 1 + … with nowrap|1 − 1 + 1 − 1 + ….

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

: $egin{array}{rcl}c_n & = &displaystyle sum_{k=0}^n a_k b_{n-k}=sum_{k=0}^n (-1)^k (-1)^{n-k} \ [1em] & = &displaystyle sum_{k=0}^n (-1)^n = (-1)^n(n+1).end{array}$

The product series is then: $sum_{n=0}^infty(-1)^n(n+1) = 1-2+3-4+cdots.$

Thus a summation method that respects the Cauchy product of two series and sums nowrap|1=1 − 1 + 1 − 1 + … = frac|1|2, will also sum nowrap|1=1 − 2 + 3 − 4 + … = frac|1|4. With the result of the previous section, this implies an equivalence between summability of nowrap|1 − 1 + 1 − 1 + … and nowrap|1 − 2 + 3 − 4 + … with methods that are linear, stable, and respect the Cauchy product.

Cesàro's theorem is a subtle example. The series nowrap|1=1 − 1 + 1 − 1 + … is Cesàro-summable in the weakest sense, called nowrap|(C, 1)-summable, while nowrap|1=1 − 2 + 3 − 4 + … requires a stronger form of Cesàro's theorem [Hardy, p.3; Weidlich, pp.52–55.] , being nowrap|(C, 2)-summable. Since all forms of Cesàro's theorem are linear and stable, the values of the sums are as we have calculated.

pecific 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

:1, −1, 2, −2, 3, −3, …,

and the arithmetic means of these partial sums are

:1, 0, frac|2|3, 0, frac|3|5, 0, frac|4|7, ….

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 frac|1|2, while the odd means are all equal to 0, so the means "of" the means converge to the average of 0 and frac|1|2, namely frac|1|4. [Hardy, p.9. For the full details of the calculation, see Weidlich, pp.17–18.] So nowrap|1 − 2 + 3 − 4 + … is (H, 2) summable to frac|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; nowrap|1 − 2 + 3 − 4 + … was his first example. [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.] The fact that frac|1|4 is the (H, 2) sum of nowrap|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 nowrap|1 − 2 + 3 − 4 + …, to frac|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. [Ferraro, pp.123–128.]

Abel summation

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

Euler proposed a generalization of the word "sum" several times; see "Euler on infinite series". In the case of nowrap|1 − 2 + 3 − 4 + …, his ideas are similar to what is now known as Abel summation:

There are many ways to see that, at least for absolute values |"x"| < 1, Euler is right in that: $1-2x+3x^2-4x^3+cdots = frac{1}{(1+x)^2}.$ One can take the Taylor expansion of the right-hand side, or apply the formal long division process for polynomials. Starting from the left-hand side, one can follow the general heuristics above and try multiplying by (1+"x") twice or squaring the geometric series nowrap|1 − "x" + "x"² − …. Euler also seems to suggest differentiating the latter series term by term. [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. [http://math.ucr.edu/home/baez/qg-winter2004/zeta.pdf Euler's Proof That 1 + 2 + 3 + … = 1/12 (PDF).] math.ucr.edu (December 19, 2003). Retrieved on March 11, 2007.]

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

: $lim_{x
ightarrow 1^{-sum_{n=1}^infty n(-x)^{n-1} = lim_{x
ightarrow 1^{-frac{1}{(1+x)^2} = frac14.$

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 nowrap|1, 2, 3, 4, …. The first element of this sequence is labeled "a"₀.

Next one needs the sequence of forward differences among nowrap|1, 2, 3, 4, …; this is just nowrap|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 nowrap|1, 1, 1, 1, … are 0. The Euler transform of nowrap|1 − 2 + 3 − 4 + … is then defined as

: $frac12 a_0-frac14Delta a_0 +frac18Delta^2 a_0 -cdots = frac12-frac14.$

In modern terminology, one says that nowrap|1 − 2 + 3 − 4 + … is Euler summable to frac|1|4.

The Euler summability implies another kind of summability as well. Representing nowrap|1 − 2 + 3 − 4 + … as

: $sum_{k=0}^infty a_k = sum_{k=0}^infty(-1)^k(k+1),$

one has the related everywhere-convergent series

: $a(x) = sum_{k=0}^inftyfrac{(-1)^k(k+1)x^k}{k!} = e^{-x}(1-x).$

The Borel sum of 1 − 2 + 3 − 4 + … is therefore [Weidlich p. 59]

: $int_0^infty e^{-x}a(x),dx = int_0^infty e^{-2x}(1-x),dx = frac12-frac14.$

eparation of scales

Saichev and Woyczyński arrive at nowrap|1=1 − 2 + 3 − 4 + … = frac|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 frac|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 [Saichev and Woyczyński, pp.260–264.]

:: $lim_{deltadownarrow0}sum_{m=0}^infty (-1)^m(m+1)varphi(delta m) = frac14.$

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 nowrap|1 − 1 + 1 − 1 + … applies the mean value theorem, but here one needs the stronger Lagrange form of Taylor's theorem.

Generalizations

Another generalization of 1 − 2 + 3 − 4 + … in a slightly different direction is the series nowrap|1 − 2^"n" + 3^"n" − 4^"n" + … for other values of "n". For positive integers "n", these series have the following Abel sums: [Knopp, p.491; there appears to be an error at this point in Hardy, p.3.] : $1-2^{n}+3^{n}-cdots = frac{2^{n+1}-1}{n+1}B_{n+1}$ where "B"_"n" are the Bernoulli numbers. For even "n", this reduces to

: $1-2^{2k}+3^{2k}-cdots = 0.$

This last 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. 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^"n" + 3^"n" − 4^"n" + etc.

:where "n" is a positive number. Here's something to laugh at, friends." [Grattan-Guinness, p.80. See Markushevich, p.48, for a different translation from the original French; the tone remains the same.]

Cesàro's teacher, Eugène Charles Catalan, also disparaged divergent series. Under Catalan's influence, Cesàro initially referred to the "conventional formulas" for nowrap|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. [Ferraro, pp.120–128.]

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 nowrap|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. [Euler et al, pp.20–25.] For example, the counterpart of nowrap|1 − 2 + 3 − 4 + … in the zeta function is the non-alternating series nowrap|1 + 2 + 3 + 4 + …, which has deep applications in modern physics but requires much stronger methods to sum.

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