# 1 − 2 + 3 − 4 + · · ·

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Prvih tisoč členov in delne vsote vrste 1 − 2 + 3 − 4 + · · ·

1 − 2 + 3 − 4 + · · · je neskončna vrsta, katere členi so zaporedna cela števila z alternirajočimi predznaki. S sumacijskim znakom lahko vsoto prvih m členov vrste zapišemo kot:

  .

Vrsta je divergentnost|divergentna, kar pomeni, da zaporedje delnih vsot (1, −1, 2, −2, …) ne konvergira proti končni limiti. Prav tako 1 − 2 + 3 − 4 + · · · nima vsote.

Kljub temu je Leonhard Euler sredi 18. stoletja zapisal enačbo (za katero je priznal, da je paradoksalna):

Strog matematični dokaz za to trditev se je pojavil šele precej kasneje. Okrog leta 1890 so Ernesto Cesàro, Émile Borel in drugi začeli raziskovati metode za določitev vsot divergentnim vrstam. Več teh metod zlahka pripiše vrsti 1 − 2 + 3 − 4 + · · · "vsoto" 1⁄4. Cesàrova vsota je ena redkih metod, ki vrste 1 − 2 + 3 − 4 + · · · ne sešteje, zato je vrsta zgled, kjer je treba uporabiti močnejšo metodo, Abelovo vsoto.

Vrsta 1 − 2 + 3 − 4 + · · · je sorodna Grandijevi vrsti 1 − 1 + 1 − 1 + · · ·. Euler ju je obavnaval kot posebna primera vrste 1 − 2n + 3n − 4n + · · · za poljuben n. Raziskave so s časom pripeljale do funkcij, ki ju danes poznamo kot Riemannovo zeta funkcijo in Dirichletovo eta funkcijo.

## [uredi] Divergenca

Členi vrste (1, −2, 3, −4, …) se ne približujejo 0, zato 1 − 2 + 3 − 4 + · · · divergira po kriteriju s členi. Za kasnejšo analizo bo koristno videti divergenco na osnovnem nivoju. Po definiciji je divergenca ali konvergenca neskončne vrste podana kot divergenca ali konvergenca zaporedja delnih vsot. Delne vsote vrste 1 − 2 + 3 − 4 + · · · so:[1]

  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,
  …

Zanimivo je opaziti, da zaporedje zavzame vsako celo število natanko enkrat (tudi ničlo, če jo štejemo kot ničto delno vsoto) in tako pokaže, da je množica celih števil števna.[2] Očitno se ne ustali pri nobenem specifičnem številu, zato vrsta 1 − 2 + 3 − 4 + · · · divergira.

## [uredi] Hevrističen pristop k seštevanju

Najenostavnejši pristop za povezavo vrste 1 − 2 + 3 − 4 + · · · z vrednostjo 1⁄4 je uporaba ugotovite, ki jih dobimo pri analizi vrste 1 − 1 + 1 − 1 + · · ·.

### [uredi] Stabilnost in linearnost

ker členi 1, −2, 3, −4, 5, −6… sledijo enostavnemu vzorcu, se lahko vrsto 1 − 2 + 3 − 4 + · · · izrazi samo s sabo in iz enačbe dobi numerično vrednost. Predpostavimo za trenutek, da se da vsoto zapisati kot s = 1 − 2 + 3 − 4 + · · · za neko število s. Pokazali hočemo, da je s = 1⁄4:

Dodajanje štirih kopij vrste 1 − 2 + 3 − 4 + · · ·, s tem da uporabljamo samo premike in dodajanje šlenov, da vsoto 1.

  s | = 1 − 2 + 3 − 4 + · · ·
  | = (1 − 1 + 1 − 1 + · · · ) + (0 − 1 + 2 − 3 + · · · )
  | = h − s,

kjer je h "vsota" vrste

  h | = 1 − 1 + 1 − 1 + · · ·
  | = 1 − (1 − 1 + 1 − · · · )
  | = 1 − h.

Če rešimo enačbi h = 1 − h in s = h − s, dobimo h = 1⁄2 in s = (1⁄2)h = 1⁄4.[3] Enako lahko enačbe preuredimo tako, da dajo(s + s) + (s + s) = h + h = 1, iz česar spet sledi s = 1⁄4; ta oblika je upodobljena na sliki.

Čeprav vrsta 1 − 2 + 3 − 4 + · · · nima klasične vsote, lahko enačbi s = 1 − 2 + 3 − 4 + · · · = 1⁄4 damo drug pomen. A generalized definition of the "sum" of a divergent series is called a summation method or summability method; there are many different methods, some of which are carried out below. What the above manipulations actually prove is the following: Given any summability method that is both linear and stable, if it sums both 1 − 1 + 1 − 1 + · · · and 1 − 2 + 3 − 4 + · · · then the sums must be 1⁄2 and 1⁄4, respectively.[4]

The above approach constrains the possible values of generalized sums of 1 − 2 + 3 − 4 + · · ·, but it does not reveal which methods will or won't sum the series in the first place. In fact, some linear and stable summability methods, such as ordinary summation, do not sum 1 − 2 + 3 − 4 + · · ·. A different way of looking at the series helps determine which methods sum it to 1⁄4: expressing it as a product.

### [uredi] Cauchyjev produkt

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, 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 1 − 2 + 3 − 4 + · · · is the Cauchy product of 1 − 1 + 1 − 1 + · · · with 1 − 1 + 1 − 1 + · · ·.

1 − 2 + 3 − 4 + · · · as twofold Cauchy product of 1 − 1 + 1 − 1 + · · ·

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

The product series is then

Many summation methods "respect" the Cauchy product of two series in one way or another. For those methods, where 1 − 1 + 1 − 1 + · · · has a generalized sum of 1⁄2, the corresponding generalized sum of 1 − 2 + 3 − 4 + · · · is (1⁄2)2 = 1⁄4. Cesàro's theorem is a subtle example; since 1 − 1 + 1 − 1 + · · · is (C, 1) summable to 1⁄2, 1 − 2 + 3 − 4 + · · · is (C, 3) summable to 1⁄4.[6] In fact, one can do slightly better: 1 − 2 + 3 − 4 + · · · is (C, 2) summable as well, although it is not (C, 1) summable itself.

## [uredi] Posamezne metode

### [uredi] Cesàro in Hölder

Data about the (H, 2) sum of 1/4

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, 2⁄3, 0, 3⁄5, 0, 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 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.[7] 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.[8] 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.[9]

### [uredi] Abelova sumacija

Some partials of 1−2x+3x2+···; 1/(1 + x)2; and limits at 1

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

Predloga:Blockquote

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

Predloga:Blockquote

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

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 1 − x + x2 − · · ·. Euler also seems to suggest differentiating the latter series term by term.[12]

In the modern view, the series 1 − 2x + 3x2 − 4x3 + · · · 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:

### [uredi] Euler in Borel

Euler summation to 1⁄2 − 1⁄4

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 a0.

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 Δa0. 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 implies another kind of summability as well. Representing 1 − 2 + 3 − 4 + · · · as

one has the related everywhere-convergent series

The Borel sum of 1 − 2 + 3 − 4 + · · · is therefore[13]

### [uredi] Ločitev skal

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 xφ(x) at +∞ are both 0, then[14]

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.

## [uredi] Generalizacija

Euler sums similar series in the 1755 Institutiones

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.[15] 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 − 2n + 3n − 4n + · · · for other values of n. For positive integers n, these series have the following Abel sums:[16]

where Bn are the Bernoulli numbers. For even n, this reduces to

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 − 2n + 3n − 4n + etc.
  where n is a positive number. Here's something to laugh at, friends."[17]

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.[18]

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.[19] 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.

## [uredi] Opombe

1. ^ Hardy p.8
2. ^ Beals p.23
3. ^ Hardy (p.6) presents these derivations with an extra step for s.
4. ^ Hardy, p.6.
5. ^ Ferraro, p.130.
6. ^ Hardy, p.3; Weidlich, pp.52–55.
7. ^ Hardy, p.9. For the full details of the calculation, see Weidlich, pp.17–18.
8. ^ 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.
9. ^ Ferraro, pp.123–128.
10. ^ Euler et al, p.2. Although the paper was written in 1749, it was not published until 1768.
11. ^ Euler et al, pp.3, 25.
12. ^ 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). math.ucr.edu (December 19 2003). Retrieved on March 11 2007.
13. ^ Weidlich p.59
14. ^ Saichev and Woyczyński, pp.260–264.
15. ^ Kline, p.313.
16. ^ Knopp, p.491; there appears to be an error at this point in Hardy, p.3.
17. ^ Grattan-Guinness, p.80. See Markushevich, p.48, for a different translation from the original French; the tone remains the same.
18. ^ Ferraro, pp.120–128.
19. ^ Euler et al, pp.20–25.

## [uredi] Viri

- Beals, Richard (2004). Analysis: an introduction. Cambridge UP. ISBN 0-521-60047-2.
- Davis, Harry F. (May 1989). Fourier Series and Orthogonal Functions. Dover. ISBN 0-486-65973-9.
- Euler, Leonhard; Lucas Willis; and Thomas J Osler (2006). Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series. The Euler Archive. Pridobljeno dne 2007-03-22. Originally published as Euler, Leonhard (1768). "Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques". Memoires de l'academie des sciences de Berlin 17: 83–106.
- Ferraro, Giovanni (June 1999). "The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics". Archive for History of Exact Sciences 54 (2): 101–135. Predloga:Doi.
- Grattan-Guinness, Ivor (1970). The development of the foundations of mathematical analysis from Euler to Riemann. MIT Press. ISBN 0-262-07034-0.
- Hardy, G.H. (1949). Divergent Series. Clarendon Press. Predloga:LCCN.
- Kline, Morris (November 1983). "Euler and Infinite Series". Mathematics Magazine 56 (5): 307–314.
- Lavine, Shaughan (1994). Understanding the Infinite. Harvard UP. ISBN 0674920961.
- Markushevich, A.I. (1967). Series: fundamental concepts with historical exposition, English translation of 3rd revised edition (1961) in Russian, Hindustan Pub. Corp.. Predloga:LCCN.
- Saichev, A.I., and W.A. Woyczyński (1996). Distributions in the physical and engineering sciences, Volume 1. Birkhaüser. ISBN 0-8176-3924-1.
- Tucciarone, John (January 1973). "The development of the theory of summable divergent series from 1880 to 1925". Archive for History of Exact Sciences 10 (1-2): 1–40. Predloga:Doi.
- Vretblad, Anders (2003). Fourier Analysis and Its Applications. Springer. ISBN 0387008365.
- Weidlich, John E. (June 1950). Summability methods for divergent series. Stanford M.S. theses. Predloga:OCLC.

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