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THIS SUM is called a DIVERGENT INFINITE SERIES (the usual sum of such a series does not exist). For such infinite divergent series, the arithmetic rules are partially violated.
The sum of such series is obtained by various special methods.
The equal sign in this expression does not have the same meaning as in 2+3=5.
This is a different math – it has added rules. And this means that it should be read like this – “equal in some sense” or even worse “it is equal in some unusual sense”
THE QUESTION is erroneous – this is not a simple arithmetic sum,
a generalized sum of a divergent series, a new generalized arithmetic.
Here, classical arithmetic is only partially observed.
I finally came to the conclusion that the theory of divergent series slightly contradicts the theory of the function of a complex variable.
In complex analysis, there is the analytic continuation theorem, which says that if an analytic function 1 is defined by a series 1 that converges in domain 1 and diverges in some domain 2 of the complex plane, then if you can find another series 2 (i.e., another analytic function 2) that will have the same values as the series 1 in domain 1 and will converge in all or part of domain 2, then such a series 2 can be considered an analytical continuation of function 1 to domain 2.
Here lies a contradiction that manifests itself explicitly in the theory of divergent series.
It is necessary to answer once and always, what is the analytical continuation of function 1 or function 2?
The theorem does not give a clear designation, and in the theory of divergent series, concepts are completely replaced by the words “we can put a certain number in correspondence with a divergent series”, in other words, it is assumed that no function 2 exists, but only function 1 and its analytical continuation exist.
So I am deeply convinced, as a philosopher, that the analytic continuation is a function 2, because it is defined by another series 2, which converges in the domain 2 and in the domain 1.
The tragedy of the situation is that mathematicians do not understand that the analytic continuation is a new function that is obtained by summation methods, i.e., by methods of converting one sequence 1 into another sequence 2, which converges equally in both domain 1 and domain 2.
If we have found a method that gives us a different sequence 2, then it also gives us a different function 2, and not the same function 1, which, according to the tradition since the time of Euler, is “assigned” a value, then traditionally follows the argument about the equal sign, which must be understood in a different non-traditional sense.
This is where we step on the rake that the sum of an infinite natural series is “equal” in some other sense to a negative number.
There is no other meaning, there is another function 2 and incorrect mathematical notation.
If we got function 2, then we don't need to say that we are still working with function 1, we need to accept once and for all that the analytical continuation is not function 1, but a new function 2, which we got by transformation from function 1, which means it is a different function and it should be written in a different way, namely with the indication of this transformation, then the equality will be fair, and there will be no need to talk about any other sense of this sign, which can only have one meaning “equal”, equal to the limit to which the new convergent series 2 tends, obtained from row 1 in a specific way.
In other words, if row 1 diverges in area 2, then it is necessary to specify a transformation on the row with the entry row 1, which makes it row 2, and then you can safely put an equal sign.
This is the philosophical and mathematical meaning of analytic continuation.
PS
There is a very good example.
Academician Yuri Matiyasevich conducted a mathematical experiment to determine the coefficients of the Dirichlet series from the zeros of the Riemann Zeta function, which is known to be given by a series that absolutely converges in the right half-plane and diverges in the left half-plane (this is exactly the function that, when substituted into the Dirichlet series, gives the natural series specified in the question and (minus)1/12 is the value of the Riemann Zeta function at -1, where the Dirichlet series diverges).
So academician Yuri Matiyasevich suggested that the coefficients of 1, i.e. the classical Dirichlet series, would be obtained. The purpose of the experiment was to see how these coefficients would converge to 1 with an increase in the number of zeros of the Riemann Zeta function, for their interpolation by calculating the discriminant (the method is used to determine the coefficients of a linear combination of linearly independent functions that defines a given function in the zeros of this function, since in this case the method of a system of linear equations is not suitable, since the system in this case is homogeneous and the method gives only a trivial solution).
A very interesting experiment, but it was doomed to failure in advance, because the Dirichlet series diverges where the Riemann Zeta function has zeros.
And this was confirmed, because in the course of the experiment (lo and behold), another function was obtained – the alternating Dirichlet series, whose zeros partially coincide with the zeros of the Riemann Zeta function.
It cannot be said that the alternating Dirichlet series is an analytical extension of the Riemann Zeta function into the critical band (since the alternating series converges in the critical band), but the fact that this function was obtained, and no other, is a natural result (so there is no miracle), since the zeros of the analytical function uniquely determine this function.
To get the coefficients of the transformed Dirichlet series that converges in the left half-plane, you should use the values of the Riemann Zeta function in the left half-plane other than zero (in this case, you can use the method of a system of linear equations, since the system will be inhomogeneous and the solution will be nontrivial), then there will be a natural result, the coefficients will tend to some of the transformations of the Dirichlet series.
And it will be a different, transformed Dirichlet series and another function-an analytical continuation of the Riemann Zeta function, the coefficients of this Dirichlet series will not be equal to 1 and it will not be a signed-alternating Dirichlet series, these will be the coefficients of one of the limit methods for converting sequences.
And then we will see what the Riemann Zeta function really is, which has zeros on the critical line, and maybe the proof of the Riemann hypothesis will become obvious.
For example, this is how the Chisaro transformation methods straighten the Riemann spiral-a sequence of partial sums of a Dirichlet series
I think this is absurd. This is the same thing that one ton of apples contains-1/12 apples….. After all, you can get a ton of apples by adding 1+2+3+4+5+… So what? Suddenly it turns out that the sum of apples is not equal to a ton, but is equal to-1/12 apples…..? Does the author of this “work” even understand the absurdity of its “conclusion” ?
And the number of stars in the universe is also equal – 1/12 ?
Consider the sequence: first term½, second 1/3*(1-1/2)=1/6, third 1/5*(1-1/2-1/6)=1/15 id. The formula of the n-th term is 1/p (where p is the n – th prime number) multiplied by one minus the sum of all the previous numbers in the sequence.
Questions: Does the series converge? If so, what is the sum equal to? What is the meaning of each member of the sequence? Who has dealt with such issues?
Here lies the deep philosophical meaning of the analytical continuation of the analytical function. The law of unity and struggle of opposites manifests itself in relation to infinite series, as a way of defining analytical functions.
Indeed, the series are divided into convergent and divergent, and it turns out that an infinite series, which is given an analytical function in one domain converges and diverges in another, the Riemann zeta function is a classic example.
The Dirichlet series that is given this function converges only in the right half-plane, and in the left half – plane it diverges – the contradiction is obvious. But unity is also obvious, in order to perform an analytical continuation of the Riemann Zeta function, it is necessary to put a certain number in correspondence with the divergent series, in other words, a single function is given by an infinite series that is different in nature, but uniform by definition – this is the analytical continuation of the function.
Indeed, by changing the summation order and performing other manipulations, any sum can be obtained from a divergent series. Therefore, methods for generalized summation of divergent series must meet several requirements:
regularity – a series constructed from generalized sums must converge in the domain where the series diverges, and must converge to the same values where the series converges (this is an essential constraint that implements the unity of opposites and the uniqueness theorem of analytic continuation)
linearity – any linear combination of numbers to which a series of generalized sums converge must be equal to the number to which a series made up of linear combinations of generalized sums converges (this is usually an obvious property if the generalized sum has a simple form)
Thus, the generalized summation of divergent series is not just a mathematical trick, but the disclosure of a hidden entity in the form of a generalized sum, which is not visible while the series converges.
To understand how this can happen, let's start with a series of 1-1+1-1+1-1+1-1+……
Since its sum does not tend to any particular value, but takes two different values in turn: 1 or 0, it is considered divergent.
However, it is possible to extend the concept of summation of series to divergent ones by first taking:
S = 1-1+1-1+1-1+1-1+…
Then the same series can be written as:
1-(1-1+1-1+1-1+1-1+… = 1-S
We have the equation:
S = 1-S
S = 0,5
Now let's take this row and square it. Multiplying the series (a1+a2+a3+a4+…) by (b1+b2+b3+b4+…) gives the series
(a1b1)+(a1b2+a2b1)+(a1b3+a2b2+a3b1)+(a1b4+a2b3+a3b2+a4b1), in which the products of those elements of the multiplier series for which the sum of the indices is constant are grouped into one term.
It turns out that (1-1+1-1+1-1+1-1+…)*(1-1+1-1+1-1+1-1+…) = 1+(1*(-1)+(-1)*1)+(1*1+(-1)*(-1)+1*1)+(1*(-1)+(-1)*1+1*(-1)+(-1)*1)+… = 1-2+3-4+5-6+7-…
Thus, the sum of a natural alternating series is 1-2+3-4+5-6+7- … is equal to 0.52 = 0.25
Now let's take it one step further. Which row should be added to the natural alternating row to get the natural one?
1-2+3-4+5-6+7-8+…
+
0+4+0+8+0+12+0+16+…
______
1+2+3+4+5+6+7+8+…
But the added series is equal to the quad natural series:
0+4+0+8+0+12+0+16+… = 4(1+2+3+4+…)
So, 1-2+3-4+5-6+7-8+… = 1+2+3+4+… -4(1+2+3+4+…)= -3(1+2+3+4+…)
-3(1+2+3+4+…)=0,25
Where from
1+2+3+4+5+6+7+8+⋯=−1/12
This result was first obtained by Ramanujan. And this is not the result of sophistry and not an empty entertainment. As it turned out, the value -1/12 for the sum of all natural numbers is now used in quantum mechanics.
It is impossible to find the sum of such a series by ordinary mathematical means. This sum was obtained by a special method that has applications in its own fields, such as string theory. But within the framework of ordinary arithmetic, we can refute this fact.
1+2+3+4+… = -1/12,
0+1+2+3+… = -1 / 12, since adding 0 does not change the sum of the series.
Now subtract the second row from the first row and get
1+1+1+1+…=0
Now add 0 to it and we get a new row
0+1+1+1+…=0
Again, we subtract the lower one from the upper row and get the contradiction 1=0.
Thank you to everyone who provided proof! Very informative!
But in the process of proof, there is a small scam – they are equated with a number of 0, 4, 0, 8, 0, 12, 0, 16, … and row 4, 8, 12, 16,…
If you follow this logic, you can get a number of 1-1+1-1+1 … represent by the difference of two identical rows 1+1+1+1+1… and its sum will become 0, not 0.5.
I agree that the ideas are interesting, and may even have a physical meaning, but the evidence presented is incorrect.
It can be seen that this expression is S1+(-S2), which implies:
ES2=ES1-ES2, where ESi is the sum of all the terms of the Si series
2ES2=ES1;
ES2=ES1/2;
ES2=(1/2)/2;
ES2=1/4.
It can be seen that this expression is S2+4S3, which implies:
ES3=ES2+4ES3;
-3ES3=ES2;
ES3=ES2/(-3);
ES3=(1/4)/(-3);
ES3=-1/12.
So it turns out such an unusual result.
In fact, there are a lot of mathematical concepts hidden in your question. Here, not the sum of the natural series is -1/12, but the value of the Riemann zeta function at -1.
The Riemann zeta function can be represented as a Dirichlet series ζ(s) = Sum 1/(n^s) from n = 1 to infinity, where s� ∈ {z = x + iy / x = Re (z) > 1}.
There are also expressions that define a zeta function for all s =�x + iy. So it turns out that the zeta function at -1 is -1/12, i.e. � ζ(-1) = -1/12. Thus, a mathematical apparatus is obtained that puts real numbers in correspondence with the divergent Dirichlet series.
Eg,
ζ(0) ~=~ Sum 1/(n^0) from n = 1 to infinity ~=~ 1 + 1 + … + 1/n^0 + … ~=~ -1/2
ζ(-1) ~=~ �Sum 1/(n^(-1)) from n = 1 to infinity ~=~ 1 + 2 + … + n + … ~=~ -1/12
Come on, calm down, smart guys. Eo is just a joke. And the catch, here's the thing: Well, yes-1/12. � All (i.e. �1+2+3+…to infinity), which arose after the Great Bang, and вернется will return to the same place, i.e. to �1/12 of the mass of carbon (to one atomic mass)). Just 1/12-/12 =0.� :)))
In fact, everything is very simple and is proved by the simplest mathematical formulas. Here, the guys from Numberphile explain everything perfectly.�http://youtu.be/ATX1dDDopy0
actually, not really… the promo clearly says that the sum of all natural numbers is -1 / 12… in fact, it begins, in the context of such and such mathematics, there and then, taking this into account… I have my own math and in it the sum is zero..
Biomusor, stupid biomusor, using marketing tricks to gain traffic for yourself
Sure. I know the answer: you can foolishly rape a bull, and if instead of brains jelly, you can add up positive integers and get a negative fraction.
Scientists have not proved this, just a popular video is walking around the network, but one error crept into it. The fact is that the sum of an infinite series does not have one property of an ordinary sum, namely associativity (the sum does not change from the permutation of terms), and this is exactly what is ignored in the proof
It's a bit of a shame… Nonsense, this shouldn't happen. Any normal person, even those who are far from mathematics, especially from its schizophrenic constructions, will say that this is not so… The sum of all natural numbers is INFINITE!!!
This is what happens when you try to cross mathematics with philosophy. The sum of all natural numbers is equal to infinity. Or, -1 / 12, as” proved “by ” scientists”. Therefore, infinity is -1 / 12. Bravo!
Hint: n cannot be replaced with n in minus 8. Why on earth?! It is too early to identify anything with the” bosonic string theory”. There is no confirmation of this theory yet. So, the fairy tale is beautiful. So, scientists have not proved anything yet. By the way, this fairy tale was born more than 100 years ago. Why is it now decided to shake off the dust?!
This once again proves that mathematicians have played too much with numbers, because in principle there is nothing more simple and logical to understand than a series of natural numbers.
The most terrible thing is that this “mathematics” is the basis of modern physical theories.
Tell me what number you want to make this series equal to – and I'll try to give you calculations that will make it exactly that…
When you start doing things that you can't do – and with divergent rows, almost nothing is possible – you can get anything…
In fact, according to normal (ordinary) mathematics, the sum of a series of (roughly speaking, an infinite number of numbers) natural numbers are not equal to- (1/12), this series diverges (that is, it is not equal to a certain number). This can be explained by recalling the simplest definition of the sum of a series: it is the limit of the sums of the first n terms of the series (that is, the limit of Sn for n tending to infinity, Sn=A1+A2+A3+…+An, Ak is the kth term of the series). In the case of a series of natural numbers, the sum has no limit (infinity is not a limit, the series tends to infinity.
If there is no limit, the sum, if you work with it as with a number, can be equal to anything, simply because if this were not the case and infinity could be considered as a number, we could, for example, subtract 1 from infinity and “get” half of this infinity (infinity, infinity-1 and half-infinity are equal), and then say that infinity is equal to two, which, of course,is
To summarize: in fact, the sum of all natural numbers is, of course, infinite, which is why, by performing various arithmetic operations with it, you can get that it is equal to some natural number.
These are special methods for summing divergent series, not sum in the traditional sense. There's a lot of math here that I don't fully understand myself, but for the sake of simplicity, we can say that these are some ways to assign some final values to divergent series, which allows you to work with these divergent series. This may be necessary in different areas of mathematics and theoretical physics.�
The point is that there is a certain algorithm, or rather algorithms that allow you to uniquely assign a certain final value to a given divergent series, and these values are not only unique for each of the series (if you use the same algorithm), but also allow you to meaningfully compare these series with each other, for example, by the” speed ” of divergence, etc.
It's just that the sum of a series of natural numbers in the classical sense is infinity, but for many other series it is also infinity. For example, the sum of factorials of natural numbers. If we plot the k-th sums of k for these two series, they will be different graphs. The analytical expression for these series is different, the partial sums are different, but the sum is the same. Something's not right here. This is where the methods of an extended, non-classical understanding of the sum of series came from.