In mathematical research and practical applications, various divergent series are often involved. Mathematicians have tried to objectively assign a real or complex value to such divergent series, defined as the sum of the corresponding series. This paper introduces two of the most well-known generalized summation methods for divergent series, interprets the idea of equalization and ergodic theory behind Chesaro's summation, and gives an interesting proof of the equation.
Written by |Ding Jiu (Professor, Department of Mathematics, University of Southern Mississippi, USA).
Readers who have studied the theory of convergence of infinite series in elementary calculus may ask, "If the series diverge, how can they be summed?""Yes, their scepticism is commendable and should be strongly promoted. However, this article is going to talk about how to find the "generalized sum" of divergent series. This is an interesting and useful question, because not only in mathematics many series are unfortunately divergent, but also in physics. As mechanologist Professor Sun Bohua recently told me, solving the "Chapman-Enskog expansion" for the Boltzmann equations, which depict the statistical behavior of non-equilibrium thermodynamic systems, is a tricky problem of divergent series.
Convergence or divergence, that's the question.
Any standard calculus textbook strictly defines when to call a series "convergent," what is a "sum" for a convergent series, and when to call a series "divergent." Given an infinite series.
where an is an infinite sequence of real numbers, which is called the general term of the series. (In this article, we do not add parentheses "" or parentheses" ( to show the symbol concisely, so a represents both the nth term of the series and the sequence itself, just as f(x) represents both the value of the function in x and the function itself.) ）
Define the part and series of the given series, i.e. s is the sum of the first n terms of the series. Because the number of terms is finite, each part and s is a number that can be calculated. If the part and the sequence s converge to a number s when n tends to infinity
That is, it is called a series.
is convergent and converges to s. In this case, s is called the sum of the series, and is written.
s=
In this case,
With definite mathematical significance, it represents a real number called "series sum".
Conversely, if the part and the series Sn do not converge to a number (also called divergence) when n tends to infinity, the series is given.
It is also said to be divergent, at this time,
It's just a mixture of a bunch of mathematical symbols, which doesn't represent any number, doesn't have any mathematical meaning, let alone sum. However, the summation of infinite series is based on a reasonable definition of "sum", and since the classical definition cannot sum series, what we are looking for is whether we can supplement the definition and "summate it in a broad sense".
The "generalized summation" of divergent series first requires a reasonable definition. Rationality here naturally includes the fulfillment of two basic requirements. One is that if the series itself has converged in the usual sense, the "sum" obtained by the generalized summation method should be equal to the sum of the series in the original sense. This requirement shows that "summing in the broad sense" has the "hereditaryness" of "summing in the narrow sense". Another requirement is based on the linear nature of the traditional summation method. We know that many operations in calculus, such as limits, derivatives, integrals, etc., have linear characteristics, such as the algebraic law of derivation [af(x)+bg(x)].'=af'(x)+bg'(x)。There is a similar assertion for series: if.
And. 
If both are convergent series and c and d are constants, then the series is also convergent and has an equation.
We want the generalized summation of divergent series to maintain this property as well.
Cesaro sued for peace.
How to define the generalized summation method of divergent series that satisfies the above two reasonable conditions?A good idea is to "average", or to use a more fashionable term: "Cesaro arithmetic averaging". This method is used to deal with non-converging sequences, and the convergence or divergence of a series is, by definition, actually about the parts and sequences of a given series. So let's think about how to turn a non-converging sequence into a "converging sequence". Let's start with a simple example.
Consider the sequence a=(-1) (n-1). It is an infinite sequence of numbers alternating between 1 and -1, and of course does not converge. However, if we take the arithmetic mean of the first n terms of this series, we get.
The Chezarro arithmetic mean sequence, called the original sequence A, is written out.
So when n tends to infinity, an tends to 0. In this way, for this divergent sequence, by averaging, we obtain a convergent sequence.
In general, for a sequence an, if it corresponds to the Cassalon arithmetic mean sequence.
Convergence and convergence to the limit l, then the original sequence an is said to converge and converge to l in the sense of the Chezarro arithmetic mean. Not only does the idea of averaging mathematically help in the convergence of number sequences, but it also makes statistical physics a respectable discipline. Even with regard to the well-being and stability of human society, the policy of "the rich pay more taxes and the poor get benefits" implemented by modern countries in taxation mostly reflects the idea of benevolent egalitarianism.
Ernesto Cesáro (1859-1906) was an Italian differential geometrician. Although he wrote a book on entangular geometry, which depicted a class of fractals now known as "Chezarro curves," and several textbooks on calculus, he proposed an averaging approach to the convergence of potentially divergent sequences that would have had the greatest impact on later generations.
The reader naturally asks, if the sequence of numbers is already converging, is its series of Chezaro arithmetic mean also converging and converging to the same limitThe answer is "yes". This is a simple proposition in the theory of the limits of the sequence, and here we might as well prove it, and by the way, review the "-n" language of the limit. Set an l. Let given a positive number, there is a natural number m, such that for all natural numbers n>m, the inequality |a-l|m, there is.
Since m has been fixed, the first term at the right end of the above inequality tends to 0 when n tends to infinity, so there is a natural number n>m such that when n > n, this term is less than 2, thus.
Certification. In addition, it is obvious that the Cesaro averaging operation is linear, i.e., the Cesaro average of the sequence Can+DBN is equal to the Cesaro average of C multiplied by A plus the Cesaro average of D multiplied by B. Taking the limit again, it can be seen that the limit operation of the Chezarro average satisfies the linear property. In this way, if the Chezaro arithmetic mean method used for divergent sequences is transplanted to the generalized summation of divergent series, this method satisfies the two basic requirements of heritability and linearity mentioned above.
To sum up, we have the Chesarro generalized summation arithmetic mean method for divergent series: for a given divergent series.
If it's parts and sequences.
Convergence to the limit s in the sense of the Chezarro arithmetic mean is called the primary series.
In the sense of the Chesaro arithmetic mean there are generalized and s. From the foregoing, we know that if the series itself has converged to and s, then it also converges to a generalized sum s in the sense of the Chezaro arithmetic average. In addition, Cesaro's generalized summation arithmetic mean method is a linear generalized summation method.
To give a historically simple and well-known example of divergent series:
Its parts and sequences are s=[1-(-1)] 2. For all natural numbers k:s (2k-1)=1 and s 2k=0, so the sequence s does not converge and hence the series diverges. On the other hand, the Chesaro arithmetic mean of the parts and sequences is listed as:
Thereby an 1 2. Or, in other words, the generalized sum of the series given in the sense of the Chezarro arithmetic mean is 1 2.
The reason for the above example is that the eighteenth-century Swiss Leonhard Euler (1707-1783) also gave the conclusion that the sum is 1 2 for this series. However, the most prolific mathematician in history played with infinite series, sometimes too freely, because he occasionally took it upon himself to assign a value to power series at non-converging points, and this is how Euler's answer was derived: he knew the famous formula for summating power series (which is the equation of geometric series with a common ratio of r).
The direct result of which is |r|<1）
So he blithely substituted x=1 on both sides of the equation to get equation 1 2=1-1+1-1+1-...。 However, this is one step away from the truth. Today, every science and engineering student who has studied the theory of elementary grade numbers knows that the convergence radius of the above power series is 1, and the convergence region is only an open interval (-1, 1). Therefore, Euler used the wrong power series assignment method, and what he obtained was a generalized sum of divergent series. In fact, if he multiplies -1 by the two ends of the power series expansion as above, he gets a non-power series of function terms.
Then substituting x=1 in the same way, there is another "and" of the same constant term series
What a ridiculous "math" this is!
Poisson-Abel generalized and.
However, if Euler does not use the direct assignment method, but takes the function 1 (1+x) at the left end of the equation.
, we get a generalized sum in another sense that is the same as the result of Cesaro's generalized summation arithmetic averaging.
Generalizing this method, we get the second classical generalized summation of divergent series: for a given divergent series.
Write the corresponding power series formally.
Suppose this series satisfies inequality 0 with respect to
Then this value s is called the generalized sum of a given series in the sense of Poisson-Abelian power series. This method was used by the French mathematician Siméon Denis Poisson (1781-1840) in the particular case of trigonometric series, but his ideas were rooted in the master of the theorem in the next paragraph, the great but premature Norwegian mathematician Niels Henrik Abel (1802-1829), so it was appropriate for them to share the naming honor.
Based on the definition statement of "generalized sum" in the sense of power series and function limits, we are reminded of Abel's theorem in calculus about the properties of power series at the endpoints of the convergence interval: assume power series.
The radius of convergence is r>0. If.
convergence, then the power series converges consistently over the closed interval [0, r]. "The series of function items.
Convergence to and function f(x)" on the set of points a is much stronger than it converges point by point to f(x) on a, which is defined as:
In a, a point x converges to the number f(x), which means that for any positive number, there is a natural number n=n(x, ) such that when n > n, it can be seen that the natural number n in the point-by-point convergence definition depends not only on , but also on x. As for consistent convergence, in its definition, the value of the natural number n does not depend on the point x:
Converges consistently to f(x) on a, and for any positive number , there is a natural number n=n( ) such that when n > n, for all x's in a, there is
Since consistent convergence is an important concept in mathematical analysis, I will give an example of a series with inconsistent convergence, or ask our old friend geometric series.
Help. The series converges everywhere on a=[0, 1). If it converges consistently on a to and function 1 (1-x), then for a concrete positive number =1, there is a natural number n that makes the inequality.
All x's in [0, 1) are true. Elementary algebra, however, reminds us that as long as.
Yes, there is. This leads to contradictions. Thus the geometric series is not uniformly convergent on [0, 1).
If some readers have difficulty understanding the previous paragraph, they can imagine "non-uniform convergence" in the following way: imagine a group of heroes and the same group of horses galloping to a destination ten miles away at the same time. Sooner or later, these men and horses would reach the finish line, but the horses were far behind the best runners. If the race is viewed as a sequence of functions, then each runner is "convergent", but the huge speed difference between horses and people leads to "not so uniform" speed to reach the finish line.
One of the nice benefits of consistent convergence on a is that as long as each function in the series of terms is continuous on a, then the sum function of the series must also be continuous on a. Going back to the conclusion of Abel's theorem, because each term in a power series is a power function, it is naturally continuous everywhere, so as long as .
convergence, then the power series.
The sum function f(x) is continuous on [0, r], especially there.
In this way, if the progression.
has converged to a real number s, then the power series.
Convergence at x=1, so it follows from Abery's theorem that it converges consistently to a continuous sum function f(x) in the closed interval [0, 1], and thus has a limit.
This shows that the Poisson-Abelian generalized summation method based on power series is hereditary. Its linear nature comes from the linear nature of sequences, series, and limits about algebraic operations. So we have a second generalized summation that satisfies the basic requirements of heritability and linearity.
Between the yoga and the bright. So, is there a relationship between Cesaro's generalized summation arithmetic mean method and Poisson-Abel's generalized summation power series?Yes. The basic relationship between them is that if the divergent series can be summed in the former generalization, then the latter can also be used, and the two generalizations are equal. This result is called the Fróbenius theorem, and it is proved as follows:
to a given series.
The Cesaro arithmetic mean sequence consists of the assumptions, its parts and sequences.
an=converges to the number s, so for >0 given by any, there exists the natural number n such that when n > n, |a-s|
The above formula is introduced. The latter also guarantees a power series.
converges to a function f(x) within the open interval (0, 1). Since the sequence A is bounded and power series.
The convergence radius is 1, and the series can be known.
To 0, on the other hand, to geometric progressions.
Item-by-item.
So there is x = to sum up, every x in the folio interval (0, 1), we have.
Since n is a positive integer that has been taken and s is a constant, the first and third terms at the right end of the last inequality above tend to 0 when x 1 -, so there is a δ> such that when 0<1-x
Since is arbitrarily positive, this proves that.
In fact, the Poisson-Abel generalized summation power series method is stronger than the Chezarro generalized summation arithmetic mean method. To illustrate this, a simple example is given. Consider the obvious divergent series (because its series of general terms does not tend to 0, which is contrary to the necessary condition for series convergence: if series.
convergence, then the general term series a 0. ）
Because. does not tend to 0, and the necessary condition for the success of Cesaro's generalized summation arithmetic mean method (1) does not hold, so this method does not apply. But on the other hand, due to power series.
There is a sum in the interval (0, 1).
It approaches the limit 1 4 when x 1, so this number is the generalized sum of the divergent series of the above constant terms in the sense of the Poisson-Abelian power series.
Finding the generalization of the Fourier series at the divergence point can better reflect the superiority of the Poisson-Abel method over the Cesaro method. Let f(x) be a periodic function with a period of 2, and its absolute function can be integrable over any bounded interval. Consider its Fourier series.
Where,;a and b are the Fourier coefficients of the function f(x). When x is fixed, the Poisson-Abelian generalized summation method is applied to this series. To do this, we establish a power series about the variable r (because the letter x is used somewhere else in this case).
One of the fundamental properties of the Fourier coefficient is that a n and b n both tend to 0 when n tends to infinity, so the "coefficient sequence" of the above power series a ncos nx + b nsin nx is consistent and bounded, which leads to the series when 0
Reuse algebraic identities.
This gives the integral expression f(x, r).
The algebraic identity used above can be simplified by multiplying the left end by the denominator at the right end, and then using the sum difference product formula in trigonometry, but this method is cumbersome. It can be abbreviated by plural numbers: order.
Then the left end of the equation to be proved is plural.
of the real part, and because.
The equation is proved, and another identity is proved.
Equation (3) This type of integral is customarily called a "Poisson integral" before the concept of "generalized summation" emerged. Poisson has already studied series (2) and the "Poisson kernel."
of the above points. It was the German mathematician Hermann Schwarz (1843-1921) who rigorously demonstrated the convergence of the following Fourier series under the Poisson-Abel generalized summation method: if f has a right limit f(x) and a left limit f(x) at point x, then.
In particular, if f is continuous at point x, this limit is equal to f(x).
Averaging and Ergodic Theory.
However, it may be an incorrect impression to think that the Poisson-Abel generalized summation power series is more favored in analytic mathematics because it is stronger than the Cessarro generalized summation arithmetic averaging. In fact, traversal theory, a comprehensive discipline in modern mathematics, is fundamentally about the study of averages. The various "ergodic theorems", to put it bluntly, are the study of the convergent properties of different kinds of "operator sequences" in the sense of the Chezarro arithmetic average.
Scholars of functional analysis would probably think that Nelson Dunford (1906-1986), a master analyst at Yale University, and his student Jacob TSchwartz, 1930-2009) co-authored the classic Linear Operators Part I: General Theory, which is the "biblical work" of this branch of pure mathematics. In the 1988-89 academic year, when I was studying the course "Ergodic Theory on [0, 1]" taught by my Ph.D. supervisor Li Tianyan (1945-2020) in the Department of Mathematics at Michigan State University, I heard a comment from him: "This book is essentially about ergodic theory. "At that time, I had just finished the academic year course "Advanced Functional Analysis" taught by Professor Sheldon Axler (1949-), and the main reference books I used included John B., a famous scholar of functional analysisConway, 1939-). Although I learned a lot from Professor Axler's beautiful lectures for three quarters, I didn't smell the traversal theory in the classroom at all. After taking Prof. Li's highly engaging lectures, my research interests shifted from optimization theory to ergodic theory. In order to confirm that what my mentor "said is true", and also to lay a solid foundation for myself to "enter the role" as soon as possible, I began to read "Linear Operator I", which I had not turned before. The book is 730 pages long, and the last chapter is titled "Applications" and is about traversal theory, while the first seven chapters are actually "prerequisites" in its service.
Starting from von Neumann's average ergodic theorem and Burkhoff's point-by-point ergodic theorem in the early thirties of the last century, numerous ergodic theorems have emerged in the past hundred years. As a representative, I will cite only one ergodemic theorem about matrices, because readers who have studied elementary linear algebra can understand it. It is assumed that the maximum absolute value of all eigenvalues of the m matrix s is 1. Just as the positive power sequence e inx of the unit complex number e ix almost never converges (the reader can make x = 4 try to see what happens, and then check if the corresponding Chesaro arithmetic mean sequence has a limit), the positive power sequence s n of the matrix s generally cannot be expected to converge unless s satisfies other properties, such as that its elements are all positive. However, as long as the power sequence s n is unanimously bounded, it's the Chezarro arithmetic mean sequence.
When n tends to infinity, it converges to a matrix p. This limit matrix p satisfies the equations p=p and sp=ps=p, so it is a projection matrix that is projected along the value space of the matrix i-s onto the zero space of i-s, where i is the unity matrix of the same order.
Just take the permutation matrix s≠i of 2 2 to fully understand the above results. Since s=i, it is clear that the odd power of s is equal to itself, and the even power is equal to the identity matrix, so the power sequence s of the matrix does not converge. On the other hand, simple calculations give that the Chezaro arithmetic mean sequence a for the sequence is equal to when n is odd.
Equal when n is even.
So when n tends to infinity. It is not difficult to verify that p 2=p and sp=ps=p, the value space of the projection matrix p is.
The zero space of p, while the zero space of p is the value space of i-s.
Incredible equation.
Looking back at the history of summation of infinite series, in the eighteenth century, Euler, who had made great contributions to the development of the various parts of calculus, was sometimes impatient to test whether the series converged or not, and other mathematicians of his time made extensive use of divergent series regardless of the consequences. One of the main reasons is that Euler holds the view that any divergent series should have a natural sum, but does not give a clear connotation of the sum of convergent series. The problem of the precise meaning of the convergence of infinite series was solved by the nineteenth-century Cauchy (1789-1857), who gave a strict mathematical definition of series convergence. Then, for decades, divergent series were excluded by analysts because of "nothing" and "nothing", and it seemed that they were not qualified to enter the elegant hall of mathematics. In 1886, Henri Poincaré (1854-1912) studied the so-called "asymptotic series" and divergent series returned. Then, in 1890, Cesaro formally defined the summation of certain divergent series, known today as the generalized summation arithmetic mean method that bears his name, although it had been implicitly used by Ferdinand Georg Frobenius (1849-1917) a decade earlier. Nowadays, the method of summation of divergent series has become a science, in addition to the two most famous methods introduced in this article, there are other generalized summaries of divergent series defined for different situations and purposes, such as Riemann summation, Herder summation, Ramanujan summation, etc.
Now that we have mentioned the legendary Indian mathematical genius Srinivasa Ramanujan (1887-1920), let's roughly explain why his summation method yielded this jaw-dropping result: 1+2+3+4+...=-1 12 as the finale of this article. The series to the left of this strange sum is generally understood to diverge to positive infinity, and it cannot be summed generally by either Cesaro's arithmetic mean or Poisson-Abel's power series. However, in his second letter to the British mathematician Godfrey Harold (1977-1947) on February 27, 1913, Ramanujan told him of this incredible equation. The letter is written:
It is with great pleasure that I read your letter of February 8, 1913, dear sir. I've been waiting for a reply from you, similar to a math professor in London who wrote to me asking me to take a closer look at Bromwich's infinite series and not fall into the trap of divergent series. ......I told him that, according to my theory, the sum of the infinite terms of the series: 1 + 2 + 3 + 4 + = 1 12. If I told you this, you would immediately say that the madhouse was my home. I elaborate on this just to convince you that if I indicate in a letter what I will continue to do, you will not be able to follow my method of proof. ......
In chapter 8 of Ramanujan's famous "Notebook I", he gives two proofs of the equation he argues, the first is formalized and lacks arguments, and the second uses Bernhard Riemann's (1826-1866) - function, which is in line with rigor. They are described as follows:
Form "Proof of": Order.
s=1+2+3+4+5+6+…Multiply 4 on both sides to get.
4s=0+4+0+8+0+12+…。
Subtract the second equation by "inserting" the first equation with a number of zeros' aligned up and down here, and there you are.
s-4s=1-2+3-4+5-6+…, and the generalization of the latter series and the previous Poisson-Abelian power series are 1 4. Therefore, there is an equation of -3s=1 4, and the solution gives s=-1 12. This proof is certainly not convincing, but it inspires the following convincing proofs.
Strict proof: let Z=X+IY. Consider the Riemann-function.
When the real part of z is >1, the Dirichlet series above converges to and (z). Multiply by the two ends of the series to get.
Subtract the two formulas and there is.
The interleaved Dirichlet series at the right end defines the Dirichlet - function (z) within its convergence region. Hence the equation of functions.
It is true for all complex numbers z that make the two series converge. Although z=-1 causes the series to diverge, it can be extended analytically in the complex analysis so that the equation still holds true for the larger region of z. This region contains -1, so -3 (-1) = (-1). The analytic extension value (-1) is equal to 1-2+3-4+....The Poisson-Abel is based on the generalized sum of 1 4 based on power seriesThis can also be seen in the equation.
Let Z=1 and note that (1)=1 and (2)=12 obtain. It follows from this.
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*: Back to the basics. Edit: kcollider