Английская Википедия:1 − 2 + 3 − 4 + ⋯
Шаблон:Short description Шаблон:Pp-move
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 <math display=block>\sum_{n=1}^m n(-1)^{n-1}.</math>
The infinite series diverges, meaning that its sequence of partial sums, Шаблон: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: <math display=block>1-2+3-4+\cdots=\frac{1}{4}.</math>
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 Шаблон:Nowrap a "value" of Шаблон:Sfrac. Cesàro summation is one of the few methods that do not sum Шаблон: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 Шаблон:Nowrap. Euler treated these two as special cases of the more general sequence Шаблон:Nowrap, where Шаблон:Nowrap and Шаблон:Nowrap 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.
Divergence
The series' terms Шаблон:Nowrap do not approach 0; therefore Шаблон:Nowrap diverges by the term test. 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 Шаблон:Nowrap are:Шаблон:Sfn Шаблон:Block indent 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 Шаблон:Nowrap, so Шаблон:Nowrap diverges.
The partial sums include every integer exactly once—even 0 if one counts the empty partial sum—and thereby establishes the countability of the set <math>\mathbb{Z}</math> of integers.Шаблон:Sfn
Heuristics for summation
Stability and linearity
Since the terms Шаблон:Nowrap follow a simple pattern, the series Шаблон:Nowrap can be manipulated by shifting and term-by-term addition to yield a numerical value. If it can make sense to write Шаблон:Nowrap for some ordinary number s, the following manipulations argue for Шаблон:Nowrap[1]
<math display=block> \begin{alignat}{5} 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{alignat} </math>
So <math>s=\frac{1}{4}</math>.
Although Шаблон:Nowrap does not have a sum in the usual sense, the equation Шаблон: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 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 Шаблон:Nowrap, the sum it produces is Шаблон:Frac.Шаблон:Sfn Furthermore, since
<math display=block> \begin{alignat}{5} 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{alignat} </math>
such a method must also sum Grandi's series as Шаблон:NowrapШаблон:Sfn
Cauchy product
In 1891, Ernesto Cesàro expressed hope that divergent series would be rigorously brought into calculus, pointing out, "One already writes Шаблон:Nowrap and asserts that both the sides are equal to Шаблон:Frac."Шаблон:Sfn 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.Шаблон:Sfn The details on his summation method are below; the central idea is that Шаблон:Nowrap is the Cauchy product (discrete convolution) of Шаблон:Nowrap with Шаблон:Nowrap.
The Cauchy product of two infinite series is defined even when both of them are divergent. In the case where an = bn = (−1)n, the terms of the Cauchy product are given by the finite diagonal sums <math display=block>\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}</math>
The product series is then <math display=block>\sum_{n=0}^\infty(-1)^n(n+1) = 1-2+3-4+\cdots.</math>
Thus a summation method that respects the Cauchy product of two series — and assigns to the series Шаблон:Nowrap the sum 1/2 — will also assign to the series Шаблон:Nowrap the sum 1/4. With the result of the previous section, this implies an equivalence between summability of Шаблон:Nowrap and Шаблон:Nowrap with methods that are linear, stable, and respect the Cauchy product.
Cesàro's theorem is a subtle example. The series Шаблон:Nowrap is Cesàro-summable in the weakest sense, called Шаблон:Nowrap while Шаблон:Nowrap requires a stronger form of Cesàro's theorem,Шаблон:Sfnm being Шаблон:Nowrap Since all forms of Cesàro's theorem are linear and stable,Шаблон:Sfn the values of the sums are as calculated above.
Specific 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: Шаблон:Block indent and the arithmetic means of these partial sums are: Шаблон:Block indent
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 Шаблон:Frac, while the odd means are all equal to 0, so the means of the means converge to the average of 0 and Шаблон:Frac, namely Шаблон:Frac.[2] So Шаблон:Nowrap is (H, 2) summable to Шаблон: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; Шаблон:Nowrap was his first example.[3] The fact that Шаблон:Frac is the (H, 2) sum of Шаблон: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 Шаблон:Nowrap to Шаблон: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.Шаблон:Sfn
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 Шаблон: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 Шаблон:Nowrap, Euler is right in that <math display=block>1-2x+3x^2-4x^3+\cdots = \frac{1}{(1+x)^2}.</math> 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 Шаблон:Nowrap twice or squaring the geometric series Шаблон:Nowrap. Euler also seems to suggest differentiating the latter series term by term.[4]
In the modern view, the generating function 1 − 2x + 3x2 − 4x3 + ... does not define a function at Шаблон:Nowrap, so that value cannot simply be substituted into the resulting expression. Since the function is defined for all Шаблон:Nowrap, one can still take the limit as x approaches 1, and this is the definition of the Abel sum: <math display=block>\lim_{x\rightarrow 1^{-}}\sum_{n=1}^\infty n(-x)^{n-1} = \lim_{x\rightarrow 1^{-}}\frac{1}{(1+x)^2} = \frac14.</math>
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 Шаблон:Nowrap The first element of this sequence is labeled a0.
Next one needs the sequence of forward differences among Шаблон:Nowrap; this is just Шаблон: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 Шаблон:Nowrap are 0. The Euler transform of Шаблон:Nowrap is then defined as <math display=block>\frac12 a_0-\frac14\Delta a_0 +\frac18\Delta^2 a_0 -\cdots = \frac12-\frac14.</math>
In modern terminology, one says that Шаблон:Nowrap is Euler summable to Шаблон:Frac.
The Euler summability also implies Borel summability, with the same summation value, as it does in general.Шаблон:Sfn
Separation of scales
Saichev and Woyczyński arrive at Шаблон: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 "Шаблон:Φ-summation methods", all of which sum the series to Шаблон:Frac:
- If Шаблон: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, thenШаблон:Sfn <math display=block>\lim_{\delta\rightarrow0}\sum_{m=0}^\infty (-1)^m(m+1)\varphi(\delta m) = \frac14.</math>
This result generalizes Abel summation, which is recovered by letting Шаблон: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 Шаблон:Nowrap applies the mean value theorem, but here one needs the stronger Lagrange form of Taylor's theorem.
Generalization
The threefold Cauchy product of Шаблон:Nowrap is Шаблон:Nowrap the alternating series of triangular numbers; its Abel and Euler sum is Шаблон:Frac.Шаблон:Sfn The fourfold Cauchy product of Шаблон:Nowrap is Шаблон:Nowrap the alternating series of tetrahedral numbers, whose Abel sum is Шаблон:Frac.
Another generalization of 1 − 2 + 3 − 4 + ... in a slightly different direction is the series Шаблон:Nowrap for other values of n. For positive integers n, these series have the following Abel sums:Шаблон:Sfnm <math display=block>1-2^{n}+3^{n}-\cdots = \frac{2^{n+1}-1}{n+1}B_{n+1}</math> where Bn are the Bernoulli numbers. For even n, this reduces to <math display=block>1-2^{2k}+3^{2k}-\cdots = 0,</math> which can be interpreted as stating that negative even values of the Riemann zeta function 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 Шаблон: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.Шаблон:Sfn
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 Шаблон: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.Шаблон:Sfn For example, the counterpart of Шаблон:Nowrap in the zeta function is the non-alternating series Шаблон:Nowrap, which has deep applications in modern physics but requires much stronger methods to sum.
See also
References
Bibliography
- Шаблон:Cite book
- Шаблон:Cite book
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- Шаблон:Cite web Originally published as Шаблон:Cite journal
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- Шаблон:Cite book 2nd Ed. published by Chelsea Pub. Co., 1991. Шаблон:LCCN. Шаблон:ISBN.
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- Шаблон:Cite book Author also known as A. I. Markushevich and Alekseï Ivanovitch Markouchevitch. Also published in Boston, Mass by Heath with Шаблон:Oclc. Additionally, Шаблон:Oclc, Шаблон:Oclc.
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Шаблон:Refend
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- ↑ Шаблон:Harvnb presents this derivation in conjunction with evaluation of Grandi's series Шаблон:Nowrap.
- ↑ Шаблон:Harvnb. For the full details of the calculation, see Шаблон:Harvnb.
- ↑ Шаблон:Harvnb; Шаблон:Harvnb. 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.
- ↑ For example, Шаблон:Harvnb advocates long division but does not carry it out; Шаблон:Harvnb calculates the Cauchy product. Euler's advice is vague; see Шаблон:Harvnb. 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). Шаблон:Webarchive math.ucr.edu (December 19, 2003). Retrieved on March 11, 2007.
