1 − 2 + 3 − 4 + ⋯

Data about the (C, 2) sum of 1/4; see #Cesàro methods

In mathematics, 1 − 2 + 3 − 4 + · · · is an alternating, divergent, infinite series. Although 1 − 2 + 3 − 4 + · · · lacks a sum in the usual sense, its Abel sum is ¹⁄₄. It provides a simple example of a series that is Abel summable but not Cesàro summable.

Divergence

The convergence or divergence of an infinite series is determined by the convergence or divergence of its sequence of partial sums. 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,

…

This sequence is notable for visiting every integer once – even 0 if one counts the empty partial sum – and therefore establishing the countability of the set ℤ of integers. It certainly does not settle down and converge to a particular number, so 1 − 2 + 3 − 4 + · · · diverges.

In a 1768 report, Leonhard Euler admits the series so 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 sum a more extended meaning…"

Summation

The simplest results connecting 1 − 2 + 3 − 4 + · · · to the value ¹⁄₄ are extensions of results concerning 1 − 1 + 1 − 1 + · · ·.

Stability and linearity

Given any definition for the sum of a possibly-divergent series 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. For the first series, see Summation of Grandi's series#Stability and linearity. For the second series, one has:

${\begin{array}{rcl}s&=&\displaystyle 1-2+3-4+\cdots \\[1em]&=&\displaystyle 1-(2-3+4-\cdots )\\[1em]&=&\displaystyle 1-(1-1+1-\cdots )-(1-2+3-\cdots )\\[1em]&=&\displaystyle 1-1/2-s,\end{array}}$

so s = ¹⁄₄.

Cauchy product

The Cauchy product of two infinite series is defined even for divergent series, and 1 − 2 + 3 − 4 + · · · is the Cauchy product of 1 − 1 + 1 − 1 + · · · with itself. Since the latter series is (C, 1) summable to ¹⁄₂, 1 − 2 + 3 − 4 + · · · is (C, 3) summable is (¹⁄₂)² = ¹⁄₄.

In fact, one can do slightly better: 1 − 2 + 3 − 4 + · · · is (C, 2) summable as well, although it is not (C, 1) summable.

In 1891, Ernesto Cesàro called on the product expression of the series as an indication that divergent series were being rigorously included in calculus:

"One already writes (1 − 1 + 1 − 1 + · · ·)² = 1 − 2 + 3 − 4 + · · · and asserts that both the sides are equal to 1/4."

Cesàro/Hölder and Abel

Further information: [[:Cesàro summation and Abel summation]]

The partial sums of 1 − 2 + 3 − 4 + · · · 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. Still, the even means converge to ¹⁄₂, while the odd means converge to 0, so the means of the means converge to the average of 0 and ¹⁄₂, namely ¹⁄₄. Strictly speaking this means that ¹⁄₄ is the (H, 2) sum of 1 − 2 + 3 − 4 + · · ·, but since Cesàro and Hölder summation have been proven equivalent, it is the (C, 2) sum as well.

The fact that ¹⁄₄ is the (H, 2) sum of 1 − 2 + 3 − 4 + · · · guarantees that it is the Abel sum as well. In 1882, when Otto Hölder first proved what we now think of as the connection between Abel summation and (H, n) summation, this series was his first example.

The Abel sum can also be computed directly:

$\lim _{x\uparrow 1}\sum _{n=1}^{\infty }n(-x)^{n-1}=\lim _{x\uparrow 1}{\frac {1}{(1+x)^{2}}}={\frac {1}{4}}.$

The kernel of this idea is common to Euler's treatment of such series:

"…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:

$1-2x+3x^{2}-4x^{3}+5x^{4}-6x^{5}+\cdots$

"that arises while expanding the expression ¹⁄_(1+x)², which this series is indeed equal to after we set x = 1."

Euler and Borel

Further information: [[:Euler summation and Borel summation]]

Euler transforms of 1 − 2 + 3 − 4 + · · · and 1 − 3 + 5 − 7 + · · ·

Euler applied another technique to the series: the Euler transform, one of his own inventions. The sequence of positive terms that makes up the alternating series is 1, 2, 3, 4, …. The sequence of forward differences is just 1, 1, 1, 1, …, and all iterated forward differences are 0. So the Euler transform of 1 − 2 + 3 − 4 + · · · is

${\frac {1}{2}}a-{\frac {1}{4}}\Delta a+{\frac {1}{8}}\Delta ^{2}a-\cdots ={\frac {1}{2}}-{\frac {1}{4}}={\frac {1}{4}}.$

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

$\sum _{k=0}^{\infty }a_{k}=\sum _{k=0}^{\infty }(-1)^{k}(k+1),$

one has the related everywhere-convergent series

$a(x)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}(k+1)x^{k}}{k!}}=e^{-x}(1-x).$

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

$\int _{0}^{\infty }e^{-x}a(x)\,dx=\int _{0}^{\infty }e^{-2x}(1-x)\,dx={\frac {1}{4}}.$

φ-summation

The method of φ-summation, as it is called by Saichev and Woyczyński, arrives at 1 − 2 + 3 − 4 + · · · = ¹⁄₄ by only two physical principles: infinitesimal relaxation and separation of scales. Mathematically, this is a result that generalizes the Abel sum:

• 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

$\lim _{\delta \downarrow 0}\sum _{m=0}^{\infty }a_{m}\varphi (\delta m)={\frac {1}{4}}.$

where a_m are the terms 1, −2, 3, −4, ….

The above can be proved as an application of the Lagrange form of Taylor's theorem. The Abel sum is recovered by letting φ = exp (−x).

Generalizations

The three-fold Cauchy product of 1 − 1 + 1 − 1 + · · · is 1 − 3 + 6 − 10 + · · ·, the alternating series of triangular numbers. Its Abel sum is ¹⁄₈, and Euler found this sum in 1755 by applying what we now call the Euler transform to the series. The four-fold 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:

$1-2^{n}+3^{n}-4^{n}+\cdots ={\frac {2^{n+1}-1}{n+1}}B_{n+1}$

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

$1-2^{2k}+3^{2k}-4^{2k}+\cdots =0.$

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 − 2ⁿ + 3ⁿ − 4ⁿ + etc.

where n is a positive number. Here's something to laugh at, friends."

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

See also

• 1 + 2 + 3 + 4 + · · ·

References

• Beals, Richard (2004). Analysis: an introduction. Cambridge UP. ISBN 0-521-60047-2.
• Davis, Harry F. (1989). Fourier Series and Orthogonal Functions. Dover. ISBN 0-486-65973-9. {{cite book}}: Unknown parameter |month= ignored (help)
• Euler, Leonhard (1768). translated into English by Thomas Osler and Lucas Willis (ed.). "Remarques sur un beau rapport entre les series des puissances tant directes que reciproques". Memoires de l'academie des sciences de Berlin. 17: 83–106.
• Ferraro, Giovanni (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. doi:10.1007/s004070050036. {{cite journal}}: Unknown parameter |month= ignored (help)
• 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. LCC QA295 .H29 1967.
• Kline, Morris (1983). "Euler and Infinite Series". Mathematics Magazine. 56 (5): 307–314. {{cite journal}}: Unknown parameter |month= ignored (help)
• Markushevich, A.I. (1967). Series: fundamental concepts with historical exposition (English translation of 3rd revised edition (1961) in Russian ed.). Hindustan Pub. Corp. LCC QA295 .M333 1967.
• Saichev, A.I., and W.A. Woyczyński (1996). Distributions in the physical and engineering sciences, Volume 1. Birkhaüser. ISBN 0-8176-3924-1, LCC QA324.W69 1996.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Tucciarone, John (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. doi:10.1007/BF00343405. {{cite journal}}: Unknown parameter |month= ignored (help)
• Weidlich, John E. (1950). Summability methods for divergent series. Stanford M.S. theses. {{cite book}}: Unknown parameter |month= ignored (help)