The sum of all natural numbers: 1 + 2 + 3 + 4 + ...
The sum of all natural numbers can be written using the following number series
What is the sum of this infinite series? Before reading further, give yourself a moment to think. If you have not met with a similar series before, and the topic of numerical series is generally not too close to you, then the answer to this question will be a big surprise for you.
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This, at first glance, completely contradictory to the intuition result, however, can be rigorously proved. But before talking about the evidence, you need to make a digression and remember the basic concepts.
To begin with, the “classical” sum of a series is the limit of partial sums of a series, if it exists and is finite. Details can be found in Wikipedia and related literature. If the final limit does not exist, then the series is called divergent.
For example, the partial sum of the first k members of the number series 1 + 2 + 3 + 4 + ... is written as follows
It is easy to understand that this sum grows without limit as k tends to infinity. Consequently, the original series is divergent and, strictly speaking, does not have a sum. There are, however, many ways to assign a finite value to divergent rows.
Row 1 + 2 + 3 + 4 + ... far from the only divergent rows. Take, for example, the Grandi series
which also diverges, but it is known that the Cesà ro summation method allows us to assign a finite value of 1/2 to this series. Cesaro summation consists in operating not with partial sums of a series, but with their arithmetic means. Allowing yourself to speculate freely, it can be said that the partial sums of the Grandi series oscillate between 0 and 1, depending on which member of the series is the last in the sum (+1 or -1), hence the value 1/2, as arithmetic average of two possible values ​​of partial sums.
Another interesting example of a divergent series is the alternating series 1 - 2 + 3 - 4 + ... , the partial sums of which also oscillate. The Abel summation method allows you to assign a final value of 1/4 to this series. Note that the Abel method is a kind of development of the Cesaro summation method, so the result of 1/4 is easy to comprehend from the point of view of intuition.
It is important to note here that summation methods are not tricks that mathematics came up with to somehow cope with divergent rows. If you apply the CesĂ ro summation or the Abel method to a convergent series, the answer given by these methods is equal to the classical sum of the convergent series.
Neither the Cesà ro summation nor the Abel method, however, allow us to work with the 1 + 2 + 3 + 4 + ... series, since the arithmetic mean of partial sums, as well as the arithmetic mean arithmetic mean, diverge. In addition, if the values ​​1/2 or 1/4 can somehow be accepted and correlated with the corresponding rows, then it is difficult to associate -1/12 with the 1 + 2 + 3 + 4 + ... series, which is an infinite sequence of positive integers
There are several ways to arrive at a -1/12 result. In this note I will only briefly dwell on one of them, namely the regularization of the zeta function . We introduce the zeta function
Substituting s = -1 , we get the original number series 1 + 2 + 3 + 4 + .... Let's do a series of simple mathematical operations on this function.
Where
is Dirichlet's eta-function
With the value s = -1, this function becomes familiar to us by a series of 1 - 2 + 3 - 4 + 5 - ... whose "sum" is equal to 1/4. Now we can easily solve the equation
Interestingly, this result finds its application in physics. For example, in string theory. Referring to page 22 of the Joseph Polchinski's String Theory:
If for some people string theory is not a convincing example due to the lack of evidence for the multitude of consequences of this theory, then we can also mention that similar methods appear in quantum field theory when trying to calculate the Casimir effect .
To two times not to go, a couple more interesting examples with zeta function
For those who want to get more information on the topic, I note that I decided to write this note after translating the corresponding article to Wikipedia , where in the "Links" section you can find a lot of additional material, mainly in English.
What is the sum of this infinite series? Before reading further, give yourself a moment to think. If you have not met with a similar series before, and the topic of numerical series is generally not too close to you, then the answer to this question will be a big surprise for you.
')
This, at first glance, completely contradictory to the intuition result, however, can be rigorously proved. But before talking about the evidence, you need to make a digression and remember the basic concepts.
To begin with, the “classical” sum of a series is the limit of partial sums of a series, if it exists and is finite. Details can be found in Wikipedia and related literature. If the final limit does not exist, then the series is called divergent.
For example, the partial sum of the first k members of the number series 1 + 2 + 3 + 4 + ... is written as follows
It is easy to understand that this sum grows without limit as k tends to infinity. Consequently, the original series is divergent and, strictly speaking, does not have a sum. There are, however, many ways to assign a finite value to divergent rows.
Row 1 + 2 + 3 + 4 + ... far from the only divergent rows. Take, for example, the Grandi series
which also diverges, but it is known that the Cesà ro summation method allows us to assign a finite value of 1/2 to this series. Cesaro summation consists in operating not with partial sums of a series, but with their arithmetic means. Allowing yourself to speculate freely, it can be said that the partial sums of the Grandi series oscillate between 0 and 1, depending on which member of the series is the last in the sum (+1 or -1), hence the value 1/2, as arithmetic average of two possible values ​​of partial sums.
Another interesting example of a divergent series is the alternating series 1 - 2 + 3 - 4 + ... , the partial sums of which also oscillate. The Abel summation method allows you to assign a final value of 1/4 to this series. Note that the Abel method is a kind of development of the Cesaro summation method, so the result of 1/4 is easy to comprehend from the point of view of intuition.
It is important to note here that summation methods are not tricks that mathematics came up with to somehow cope with divergent rows. If you apply the CesĂ ro summation or the Abel method to a convergent series, the answer given by these methods is equal to the classical sum of the convergent series.
Neither the Cesà ro summation nor the Abel method, however, allow us to work with the 1 + 2 + 3 + 4 + ... series, since the arithmetic mean of partial sums, as well as the arithmetic mean arithmetic mean, diverge. In addition, if the values ​​1/2 or 1/4 can somehow be accepted and correlated with the corresponding rows, then it is difficult to associate -1/12 with the 1 + 2 + 3 + 4 + ... series, which is an infinite sequence of positive integers
There are several ways to arrive at a -1/12 result. In this note I will only briefly dwell on one of them, namely the regularization of the zeta function . We introduce the zeta function
Substituting s = -1 , we get the original number series 1 + 2 + 3 + 4 + .... Let's do a series of simple mathematical operations on this function.
Where
is Dirichlet's eta-functionWith the value s = -1, this function becomes familiar to us by a series of 1 - 2 + 3 - 4 + 5 - ... whose "sum" is equal to 1/4. Now we can easily solve the equation
Interestingly, this result finds its application in physics. For example, in string theory. Referring to page 22 of the Joseph Polchinski's String Theory:
If for some people string theory is not a convincing example due to the lack of evidence for the multitude of consequences of this theory, then we can also mention that similar methods appear in quantum field theory when trying to calculate the Casimir effect .
To two times not to go, a couple more interesting examples with zeta function
For those who want to get more information on the topic, I note that I decided to write this note after translating the corresponding article to Wikipedia , where in the "Links" section you can find a lot of additional material, mainly in English.
Source: https://habr.com/ru/post/53883/
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