1 − 2 + 3 − 4 + ⋯

The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ... The graph is situated with positive integers to the right and negative integers to the left.

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

$\sum _{n=1}^{m}n(-1)^{n-1}.$

The infinite series diverges, meaning that its sequence of partial sums, Template:Nowrap, does not tend towards any finite limit. 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 Template:Nowrap a "value" of Template:Sfrac. Cesàro summation is one of the few methods that do not sum Template:Nowrap, 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 Template:Nowrap. Euler treated these two as special cases of Template:Nowrap for arbitrary n, a line of research extending 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 Template:Nowrap 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 Template:Nowrap 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 including every integer exactly once—even 0 if one counts the empty partial sum—and thereby establishing the countability of the set $\mathbb {Z}$ of integers. The sequence of partial sums clearly shows that the series does not converge to a particular number (for any proposed limit x, we can find a point beyond which the subsequent partial sums are all outside the interval [x−1, x+1]), so Template:Nowrap diverges.

Heuristics for summation

Stability and linearity

Since the terms 1, −2, 3, −4, 5, −6, ... follow a simple pattern, the series Template:Nowrap can be manipulated by shifting and term-by-term addition to yield a numerical value. If it can make sense to write Template:Nowrap for some ordinary number s, the following manipulations argue for Template:Nowrap

${\begin{array}{rclllll}4s&=&&(1-2+3-4+\cdots )&{}+(1-2+3-4+\cdots )&{}+(1-2+3-4+\cdots )&{}+(1-2+3-4+\cdots )\\&=&&(1-2+3-4+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}+(1-2)+(3-4+5-6\cdots )\\&=&&(1-2+3-4+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}-1+(3-4+5-6\cdots )\\&=&1+&(1-2+3-4+\cdots )&{}+(-2+3-4+5+\cdots )&{}+(-2+3-4+5+\cdots )&{}+(3-4+5-6\cdots )\\&=&1+[&(1-2-2+3)&{}+(-2+3+3-4)&{}+(3-4-4+5)&{}+(-4+5+5-6)+\cdots ]\\&=&1+[&0+0+0+0+\cdots ]\\4s&=&1\end{array}}$

Adding 4 copies of Template:Nowrap using only shifts and term-by-term addition, yields 1. The left side and right side each demonstrates two copies of Template:Nowrap adding to Template:Nowrap

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 Template:Nowrap 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 (some of which are described below) and it is desirable that they share certain properties 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 Template:Nowrap, the sum it produces is Template:Frac. Furthermore, since

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

such a method must also sum Grandi's series as Template:Nowrap

Cauchy product

In 1891, Ernesto Cesàro expressed hope that divergent series would be rigorously brought into calculus, pointing out, "One already writes Template:Nowrap and asserts that both the sides are equal to Template:Frac." 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. The details on his summation method are below; the central idea is that Template:Nowrap is the Cauchy product (discrete convolution) of Template:Nowrap with Template:Nowrap.

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

${\begin{array}{rcl}c_{n}&=&\displaystyle \sum _{k=0}^{n}a_{k}b_{n-k}=\sum _{k=0}^{n}(-1)^{k}(-1)^{n-k}\\[1em]&=&\displaystyle \sum _{k=0}^{n}(-1)^{n}=(-1)^{n}(n+1).\end{array}}$

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 assigns to the series Template:Nowrap the sum 1/2 — will also assign to the series Template:Nowrap the sum 1/4. With the result of the previous section, this implies an equivalence between summability of Template:Nowrap and Template:Nowrap with methods that are linear, stable, and respect the Cauchy product.

Cesàro's theorem is a subtle example. The series Template:Nowrap is Cesàro-summable in the weakest sense, called Template:Nowrap while Template:Nowrap requires a stronger form of Cesàro's theorem, being Template:Nowrap Since all forms of Cesàro's theorem are linear and stable, the values of the sums are as we have calculated.

Specific methods

Cesàro and Hölder

Data about the (H, 2) sum of Template:Frac

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, Template:Frac, 0, Template:Frac, 0, Template:Frac, ....

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 Template:Frac, while the odd means are all equal to 0, so the means of the means converge to the average of 0 and Template:Frac, namely Template:Frac. So Template:Nowrap is (H, 2) summable to Template:Frac.

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; Template:Nowrap was his first example. The fact that Template:Frac is the (H, 2) sum of Template:Nowrap 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 Template:Nowrap to Template:Frac 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.

Abel summation

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:

Euler proposed a generalization of the word "sum" several times. In the case of Template:Nowrap, his ideas are similar to what is now known as Abel summation:

There are many ways to see that, at least for absolute values Template:Nowrap, 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 Template:Nowrap twice or squaring the geometric series Template:Nowrap. Euler also seems to suggest differentiating the latter series term by term.

In the modern view, the series 1 − 2x + 3x² − 4x³ + ... does not define a function at Template:Nowrap, so that value cannot simply be substituted into the resulting expression. Since the function is defined for all Template:Nowrap, one can still take the limit as x approaches 1, and this is the definition of the Abel sum:

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

Euler and Borel

Euler summation to Template:Frac − Template:Frac. Positive values are shown in white, negative values are shown in brown, and shifts and cancellations are shown in green.

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 Template:Nowrap The first element of this sequence is labeled a₀.

Next one needs the sequence of forward differences among Template:Nowrap; this is just Template:Nowrap 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 Template:Nowrap are 0. The Euler transform of Template:Nowrap is then defined as

${\frac {1}{2}}a_{0}-{\frac {1}{4}}\Delta a_{0}+{\frac {1}{8}}\Delta ^{2}a_{0}-\cdots ={\frac {1}{2}}-{\frac {1}{4}}.$

In modern terminology, one says that Template:Nowrap is Euler summable to Template:Frac.

The Euler summability implies another kind of summability as well. Representing Template:Nowrap as

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

one has the related everywhere-convergent series

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

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

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

Separation of scales

Saichev and Woyczyński arrive at Template:Nowrap 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 "Template:Φ-summation methods", all of which sum the series to Template:Frac:

• If Template: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

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

This result generalizes Abel summation, which is recovered by letting Template: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 Template:Nowrap applies the mean value theorem, but here one needs the stronger Lagrange form of Taylor's theorem.

Generalization

Excerpt from p.233 of the E212 — Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum. Euler sums similar series, ca. 1755.

The threefold Cauchy product of Template:Nowrap is Template:Nowrap the alternating series of triangular numbers; its Abel and Euler sum is Template:Frac. The fourfold Cauchy product of Template:Nowrap is Template:Nowrap the alternating series of tetrahedral numbers, whose Abel sum is Template:Frac.

Another generalization of 1 − 2 + 3 − 4 + ... in a slightly different direction is the series Template:Nowrap for other values of n. For positive integers n, these series have the following Abel sums:

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

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:

Cesàro's teacher, Eugène Charles Catalan, also disparaged divergent series. Under Catalan's influence, Cesàro initially referred to the "conventional formulas" for Template:Nowrap 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.

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 Template:Nowrap 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. For example, the counterpart of Template:Nowrap in the zeta function is the non-alternating series Template:Nowrap, which has deep applications in modern physics but requires much stronger methods to sum.

See also

• 1 + 2 + 3 + 4 + ⋯
• 1 + 1 + 1 + 1 + ⋯
• 1 + 2 + 4 + 8 + ·...
• 1 − 2 + 4 − 8 + ⋯

Footnotes

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