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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
s, given alternating signs. Using sigma summation notation the sum of the first ''m'' terms of the series can be expressed as \sum_^m n(-1)^. The infinite series diverges, meaning that its sequence of partial sums, , does not tend towards any finite
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. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation: 1-2+3-4+\cdots=\frac. 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 a "value" of . Cesàro summation is one of the few methods that do not sum , so the series is an example where a slightly stronger method, such as
Abel summation In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...
, is required. The series is closely related to Grandi's series . Euler treated these two as special cases of the more general sequence , where and 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 The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.


Divergence

The series' terms do not approach 0; therefore diverges by the
term test In mathematics, the ''n''th-term test for divergenceKaczor p.336 is a simple test for the divergence of an infinite series:If \lim_ a_n \neq 0 or if the limit does not exist, then \sum_^\infty a_n diverges.Many authors do not name this test or ...
. 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 are: 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 , so diverges. The partial sums include every integer exactly once—even 0 if one counts the empty partial sum—and thereby establishes the
countability In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
of the set \mathbb of integers.


Heuristics for summation


Stability and linearity

Since the terms follow a simple pattern, the series can be manipulated by shifting and term-by-term addition to yield a numerical value. If it can make sense to write for some ordinary number ''s'', the following manipulations argue for \begin 4s&= &&(1-2+3-\cdots) \ \ &&+(1-2+3-4+\cdots) && +(1-2+3-4+\cdots) &&+(1-2+3-4+\cdots) \\ &= &&(1-2+3-\cdots) && +1 +(-2+3-4+\cdots) \ \ && +1+(-2+3-4+\cdots) \ \ &&+1-2+(3-4+\cdots) \\ &=\ 1+ &&(1-2+3-\cdots) && +(-2+3-4+\cdots) && +(-2+3-4+\cdots) &&+(3-4+5-\cdots) \\ &=\ 1+ &&(1-2-2+3) && +(-2+3+3-4) && +(3-4-4+5) &&+\cdots \ \\ &=\ 1+ && 0+0+0+\cdots\ \\ 4s&=\ 1 \end So s=\frac. Although does not have a sum in the usual sense, the equation 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 , the sum it produces is . Furthermore, since \begin 2s&= &&(1-2+3-4+\cdots) \ \ &&+(1-2+3-4+5-\cdots) \\ &= && 1 +(-2+3-4+\cdots) \ \ &&+1-2+(3-4+5-\cdots) \\ &=\ 0+ &&(-2+3-4+\cdots) &&+(3-4+5-\cdots) \\ &=\ 0+ &&(-2+3) \quad +(3-4) && +(-4+5) \quad +\cdots \ \\ 2s&=\ && 1-1+1-1+\cdots \end such a method must also sum Grandi's series as


Cauchy product

In 1891, Ernesto Cesàro expressed hope that divergent series would be rigorously brought into calculus, pointing out, "One already writes and asserts that both the sides are equal to ." 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
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; the central idea is that is the Cauchy product (discrete convolution) of with . The Cauchy product of two infinite series is defined even when both of them are divergent. In the case where , the terms of the Cauchy product are given by the finite diagonal sums \begin c_n & = \sum_^n a_k b_=\sum_^n (-1)^k (-1)^ \\ & = \sum_^n (-1)^n = (-1)^n(n+1). \end The product series is then \sum_^\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 the sum 1/2 — will also assign to the series the sum 1/4. With the result of the previous section, this implies an equivalence between summability of and with methods that are linear, stable, and respect the Cauchy product. Cesàro's theorem is a subtle example. The series is Cesàro-summable in the weakest sense, called while requires a stronger form of Cesàro's theorem, being Since all forms of Cesàro's theorem are linear and stable, the values of the sums are as calculated above.


Specific methods


Cesàro and Hölder

To find the Cesàro sum of , if it exists, one needs to compute the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
s 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 is not Cesàro summable. There are two well-known generalizations of Cesàro summation: the conceptually simpler of these is the sequence of methods for natural numbers ''n''. The sum is Cesàro summation, and higher methods repeat the computation of means. Above, the even means converge to , while the odd means are all equal to 0, so the means ''of the means'' converge to the average of 0 and , namely . So is summable to . The "H" stands for
Otto Hölder Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart. Early life and education Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Christ ...
, who first proved in 1882 what mathematicians now think of as the connection between
Abel summation In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...
and summation; was his first example. The fact that is the sum of 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 methods. It has been proven that (C, ''n'') summation and summation always give the same results, but they have different historical backgrounds. In 1887, Cesàro came close to stating the definition of summation, but he gave only a few examples. In particular, he summed to by a method that may be rephrased as but was not justified as such at the time. He formally defined the methods in 1890 in order to state his theorem that the Cauchy product of a -summable series and a -summable series is -summable.


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. In the case of , his ideas are similar to what is now known as
Abel summation In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...
: There are many ways to see that, at least for
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
s , Euler is right in that 1-2x+3x^2-4x^3+\cdots = \frac. 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 twice or squaring the geometric series . Euler also seems to suggest differentiating the latter series term by term.For example, advocates long division but does not carry it out; calculates the Cauchy product. Euler's advice is vague; see .
John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
even suggests a category-theoretic method involving multiply
pointed set In mathematics, a pointed set (also based set or rooted set) is an ordered pair (X, x_0) where X is a set and x_0 is an element of X called the base point, also spelled basepoint. Maps between pointed sets (X, x_0) and (Y, y_0) – called based ma ...
s and the
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. Baez, John C
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
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
does not define a function at , so that value cannot simply be substituted into the resulting expression. Since the function is defined for all , one can still take the limit as ''x'' approaches 1, and this is the definition of the Abel sum: \lim_\sum_^\infty n(-x)^ = \lim_\frac = \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 The first element of this sequence is labeled ''a''0. Next one needs the sequence of forward differences among ; this is just 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 are 0. The Euler transform of is then defined as \frac12 a_0-\frac14\Delta a_0 +\frac18\Delta^2 a_0 -\cdots = \frac12-\frac14. In modern terminology, one says that is Euler summable to . The Euler summability also implies Borel summability, with the same summation value, as it does in general.


Separation of scales

Saichev and Woyczyński arrive at 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 "-summation methods", all of which sum the series to : * If ''φ''(''x'') is a function whose first and second derivatives are continuous and integrable over , such that and the limits of ''φ''(''x'') and ''xφ''(''x'') at +∞ are both 0, then \lim_\sum_^\infty (-1)^m(m+1)\varphi(\delta m) = \frac14. This result generalizes Abel summation, which is recovered by letting . 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 applies the
mean value theorem In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
, but here one needs the stronger Lagrange form of
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
.


Generalization

The threefold Cauchy product of is the alternating series of triangular numbers; its Abel and Euler sum is . The fourfold Cauchy product of is the alternating series of tetrahedral numbers, whose Abel sum is . Another generalization of in a slightly different direction is the series for other values of ''n''. For positive integers ''n'', these series have the following Abel sums: 1-2^+3^-\cdots = \fracB_ where ''B''''n'' are the Bernoulli numbers. For even ''n'', this reduces to 1-2^+3^-\cdots = 0, which can be interpreted as stating that negative even values of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
are zero. This 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 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 was the functional equation of the eta function, which leads directly to the functional equation of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
. 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 in the zeta function is the non-alternating series , which has deep applications in modern physics but requires much stronger methods to sum.


See also

*
1 + 2 + 3 + 4 + ⋯ 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1 ...
*
1 + 1 + 1 + 1 + ⋯ 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit (measurement), unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment ...
* 1 + 2 + 4 + 8 + ⋯ *
1 − 2 + 4 − 8 + ⋯ 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. ...


References


Bibliography

* * * * Originally published as * * * 2nd Ed. published by Chelsea Pub. Co., 1991. . . * * * * Author also known as A. I. Markushevich and Alekseï Ivanovitch Markouchevitch. Also published in Boston, Mass by Heath with . Additionally, , . * * * * * {{DEFAULTSORT:1 2 3 4 Divergent series Mathematical series Mathematical paradoxes