The sum of the divergent series. How to determine if an infinite series converges

Example 8. Given a row

Investigate its convergence at points x= 1 and X= 3, x= -2.

When x = 1, this series turns into a number series

Let us investigate the convergence of this series by the d'Alembert test. We have

Those. the series converges.

For x = 3 we get the series

Which diverges, since the necessary criterion for the convergence of the series is not satisfied

For x = -2 we get

This is an alternating series, which, according to the Leibniz test, converges.

So at the points x= 1 and X= -2. the series converges, and at the point x= 3 diverges.

Expansion of elementary functions in the Maclaurin series:

Near Taylor for function f(x) is called a power series of the form

If, a = 0, then we get a special case of the Taylor series

which is called next to Maclaurin.

A power series within its interval of convergence can be term-by-term differentiated and integrated as many times as desired, and the resulting series have the same interval of convergence as the original series.

Two power series can be added and multiplied term by term according to the rules of addition and multiplication of polynomials. In this case, the interval of convergence of the resulting new series coincides with the common part of the intervals of convergence of the original series.

To expand a function into a Maclaurin series, it is necessary:

1) calculate the values ​​of the function and its successive derivatives at the point x= 0, i.e. , , .

8. Expand the Maclaurin series of functions.

In practice, it is often not as important to find the sum of a series as to answer the question of the convergence of the series. For this purpose, convergence criteria based on the properties of the common term of the series are used.

A necessary criterion for the convergence of a series

THEOREM 1

If the rowconverges, then its common term tends to zero at
, those.
.

Briefly: if the series converges, then its common term tends to zero.

Proof. Let the series converge and its sum is equal to . For anyone partial sum

.

Then . 

From the proved necessary criterion for convergence it follows a sufficient criterion for the divergence of the series: if at
the common term of the series does not tend to zero, then the series diverges.

Example 4

For this series, the common term
and
.

Therefore, this series diverges.

Example 5 Investigate for convergence series

It is obvious that the common term of this series, the form of which is not indicated due to the cumbersome expression, tends to zero at
, i.e. the necessary criterion for the convergence of the series is satisfied, but this series diverges, since its sum tends to infinity.

Positive sign series

A series of numbers all of whose members are positive is called sign-positive.

THEOREM 2 (Criterion for the convergence of a positive series)

For a positive series to converge, it is necessary and sufficient that all its partial sums be bounded above by the same number.

Proof. Since for any
, then, i.e. sequence
- monotonically increasing, therefore, for the existence of a limit, it is necessary and sufficient to restrict the sequence from above by some number.

This theorem is more theoretical than practical. The following are other convergence criteria that are of greater use.

Sufficient conditions for the convergence of sign-positive series

THEOREM 3 (First test of comparison)

Let two positive series be given:

(1)

(2)

and, starting from some number
, for anyone
the inequality
Then:

Schematic notation of the first sign of comparison:

descent.  descent.

flowflow

Proof. 1) Since the elimination of a finite number of terms of the series does not affect its convergence, we will prove the theorem for the case
. Let for anyone
we have

, (3)

where
and
are the partial sums of series (1) and (2), respectively.

If series (2) converges, then there is a number
. Since the sequence
- increasing, its limit is greater than any of its members, i.e.
for anyone . Hence from inequality (3) it follows
. Thus, all partial sums of series (1) are bounded from above by the number . According to Theorem 2, this series converges.

2) Indeed, if series (2) converged, then series (1) would also converge by comparison. 

To apply this feature, such standard series are often used, the convergence or divergence of which is known in advance, for example:

3) - Dirichlet series (it converges at
and diverges at
).

In addition, series are often used, which can be obtained using the following obvious inequalities:

,

,
,
.

Consider, using specific examples, a scheme for studying a positive-sign series for convergence using the first comparison criterion.

Example 6 Explore a number
for convergence.

Step 1. Let's check the positive sign of the series:
for

Step 2. Let's check the fulfillment of the necessary criterion for the convergence of the series:
. Because
, then

(if calculating the limit is difficult, then this step can be skipped).

Step 3. We use the first sign of comparison. To do this, we select a standard series for this series. Because
, then as a standard we can take the series
, i.e. Dirichlet row. This series converges because the exponent
. Therefore, according to the first criterion of comparison, the series under study also converges.

Example 7 Explore a number
for convergence.

1) This series is sign-positive, since
for

2) The necessary criterion for the convergence of the series is satisfied, because

3) Let's select a series-standard. Because
, then as a standard we can take the geometric series

. This series converges, therefore, the series under study also converges.

THEOREM 4 (Second comparison test)

If for sign-positive series and there is a non-zero finite limit
, then rows converge or diverge at the same time.

Proof. Let series (2) converge; Let us prove that then the series (1) also converges. Let's pick some number , more than . From the condition
the existence of such a number that for everyone
the inequality
, or, which is the same,

(4)

Discarding in rows (1) and (2) the first terms (which does not affect the convergence), we can assume that inequality (4) is valid for all
But a series with a common term
converges due to the convergence of series (2). According to the first criterion of comparison, inequality (4) implies the convergence of series (1).

Now let series (1) converge; Let us prove the convergence of series (2). To do this, simply reverse the roles of the given rows. Because

then, by what was proved above, the convergence of series (1) should imply the convergence of series (2). 

If
at
(a necessary criterion for convergence), then from the condition
, follows that and are infinitesimals of the same order of smallness (equivalent at
). Therefore, if given a series , where
at
, then for this series we can take the standard series , where the common term has the same order of smallness as the common term of the given series.

When choosing a reference series, you can use the following table of equivalent infinitesimal for
:

1)
; 4)
;

2)
; 5)
;

3)
; 6)
.

Example 8 Investigate for convergence series

.

for anyone
.

Because
, then we take as a reference series the harmonic divergent series
. Since the limit of the ratio of common terms and is finite and different from zero (it is equal to 1), then on the basis of the second criterion of comparison, this series diverges.

Example 9
on two grounds of comparison.

This series is positive, because
, and
. Insofar as
, then the harmonic series can be taken as a reference series . This series diverges and, therefore, according to the first sign of comparison, the series under study also diverges.

Since for the given series and the reference series, the condition
(here the 1st remarkable limit is used), then based on the second comparison criterion, the series
- diverges.

THEOREM 5 (D'Alembert test)

there is a finite limit
, then the series converges at
and diverges at
.

Proof. Let
. Let's take any number , concluded between and 1:
. From the condition
it follows that starting from some number the inequality

;
;
(5)

Consider the series

According to (5), all terms of series (6) do not exceed the corresponding terms of an infinite geometric progression
Insofar as
, this progression is convergent. From here, by virtue of the first sign of comparison, the convergence of the series follows

Happening
consider for yourself.

Remarks :

it follows that the remainder of the series

.

The d'Alembert test is convenient in practice when the common term of the series contains an exponential function or a factorial.

Example 10 Investigate for convergence series according to d'Alembert.

This series is positive and

.

(Here, in the calculation, the L'Hopital rule is applied twice).

then this series converges by the d'Alembert criterion.

Example 11..

This series is positive and
. Insofar as

then the series converges.

THEOREM 6 (Cauchy test)

If for a sign-positive series there is a finite limit
, then at
the series converges, and
the row diverges.

The proof is similar to Theorem 5.

Remarks :

Example 12. Investigate for convergence series
.

This series is positive, because
for anyone
. Since the calculation of the limit
causes certain difficulties, we omit the verification of the feasibility of the necessary criterion for the convergence of the series.

then the given series diverges according to the Cauchy criterion.

THEOREM 7 (Integral test for Maclaurin-Cauchy convergence)

Let a row be given

whose terms are positive and do not increase:

Let further
is a function that is defined for all real
, is continuous, does not increase, and

Such amounts are called endless rows, and their terms are terms of the series. (An ellipsis means that the number of terms is infinite.) Solutions to complex mathematical problems can rarely be represented in an exact form using formulas. However, in most cases these solutions can be written as series. After such a solution is found, the methods of the theory of series allow us to estimate how many terms of the series must be taken for specific calculations or how to write the answer in the most convenient form. Along with numerical series, we can consider the so-called. functional rows, whose terms are functions . Many functions can be represented using function series. The study of numerical and functional series is an important part of calculus.

In examples (1) and (2) it is relatively easy to guess by what law successive terms are formed. The law of the formation of the members of a series may be much less obvious. For example, for series (3) it will become clear if this series is written in the following form:

Converging rows.

Since the addition of an infinite number of terms of a series is physically impossible, it is necessary to determine what exactly should be understood by the sum of an infinite series. It can be imagined that these addition and subtraction operations are performed sequentially, one after the other, for example, on a computer. If the resulting sums (partial sums) come closer and closer to a certain number, then it is reasonable to call this number the sum of an infinite series. Thus, the sum of an infinite series can be defined as the limit of a sequence of partial sums. Moreover, such a series is called convergent.

Finding the sum of series (3) is not difficult if you notice that the transformed series (4) can be written as

Successive partial sums of series (5) are

etc.; you can see that the partial sums tend to 1. Thus, this series converges and its sum is 1.

As an example of infinite series, consider infinite decimal fractions. So, 0.353535... is an infinite recurring decimal fraction, which is a compact way of writing the series

The law of formation of successive members is clear here. Similarly, 3.14159265... means

but the law of formation of subsequent members of the series is not obvious here: the digits form the decimal expansion of the number p, and it is difficult to immediately say what, for example, the 100,000th digit is, although theoretically this figure can be calculated.

Diverging rows.

An infinite series that does not converge is said to diverge (such a series is called divergent). For example, a row

diverges, since its partial sums are 1/2, 1, 1 1 / 2 , 2,.... These sums do not tend to any number as a limit, since by taking enough terms of the series, we can make a partial sum no matter how big. Row

also diverges, but for a different reason: the partial sums of this series alternately turn to 1, then to 0 and do not tend to the limit.

Summation.

To find the sum of a convergent series (with a given accuracy) by successively summing its terms, although theoretically possible, it is practically difficult to implement. For example, a row

converges, and its sum to within ten decimal places is 1.6449340668, but in order to calculate it with this accuracy, it would be necessary to take approx. 20 billion members. Such series are usually summarized by first transforming them using various techniques. In this case, algebraic or computational methods are used; for example, one can show that the sum of series (8) is equal to p 2 /6.

Notation.

When working with infinite series, it is useful to have convenient notation. For example, the final sum of series (8) can be written as

This entry indicates that n successively set to 1, 2, 3, 4, and 5, and the results are added up:

Similarly, series (4) can be written as

where the symbol Ґ indicates that we are dealing with an infinite series, and not with its finite part. The symbol S (sigma) is called the summation sign.

Infinite geometric progression.

We were able to sum series (4) because there was a simple formula for its partial sums. Similarly, one can find the sum of series (2), or in general,

if r takes values ​​between –1 and 1. In this case, the sum of series (9) is equal to 1/(1 – r); for other values r series (9) diverges.

You can think of periodic decimals like 0.353535... as another way of writing an infinite geometric progression.

This expression can also be written as

where series (9) with r= 0.01; therefore, the sum of series (10) is equal to

In the same way, any periodic decimal fraction can be represented as a common fraction.

Signs of convergence.

In the general case, there is no simple formula for the partial sums of an infinite series, so special methods are used to establish the convergence or divergence of the series. For example, if all terms of a series are positive, then it can be shown that the series converges if each of its terms does not exceed the corresponding term of the other series, which is known to converge. In the accepted notation, this can be written as follows: if a n i 0 and converges, then converges if 0 j b n Ј a n. For example, since series (4) converges and

then we can conclude that series (8) also converges. Comparison is the main method for establishing the convergence of many series by comparing them with the simplest convergent series. Sometimes more special convergence criteria are used (they can be found in the literature on series theory.) Here are a few more examples of convergent series with positive terms:

Comparison can also be used to establish the divergence of a series. If the series diverges, then the series also diverges if 0 J b n Ј a n.

Examples of divergent series are the series

and, in particular, since harmonic series

The divergence of this series can be verified by counting the following partial sums:

etc. Thus, the partial sums that end in the terms 1/4, 1/8, 1/16, 1/32, j exceed the partial sums of the divergent series (6), and therefore the series (14) must diverge.

Absolute and conditional convergence.

For lines like

the comparison method is not applicable, since the terms of this series have different signs. If all terms of series (15) were positive, then we would get series (3), which is known to converge. It can be shown that this also implies the convergence of series (15). When by changing the signs of the negative terms of the series to the opposite ones it can be turned into a convergent one, they say that the original series converges absolutely.

The alternating harmonic series (1) is not absolutely convergent, since series (14), consisting of the same but only positive terms, does not converge. However, with the help of special convergence criteria for alternating series, one can show that the series (1) actually converges. A convergent series that does not converge absolutely is called conditionally convergent.

Operations with rows.

Based on the definition of a convergent series, it is easy to show that its convergence is not violated by deleting or assigning a finite number of terms to it, as well as by multiplying or dividing all the terms of the series by the same number (of course, division by 0 is excluded). For any rearrangement of the terms of an absolutely convergent series, its convergence is not violated, and the sum does not change. For example, since the sum of the series (2) is 1, the sum of the series

is also equal to 1, since this series is obtained from series (2) by interchanging neighboring terms (the 1st term with the 2nd, etc.). You can arbitrarily change the order of the terms of an absolutely convergent series, as long as all the members of the original series are present in the new series. On the other hand, rearranging the terms of a conditionally convergent series can change its sum and even make it divergent. Moreover, the terms of a conditionally convergent series can always be rearranged so that it converges to any predetermined sum.

Two convergent series S a n and S b n can be added (or subtracted) term by term, so that the sum of the new series (which also converges) is added to the sums of the original series, in our notation

Under additional conditions, for example, if both series converge absolutely, they can be multiplied by each other, as is done for finite sums, and the resulting double series ( see below) will converge to the product of the sums of the original series.

Summability.

Despite the fact that our definition of the convergence of an infinite series seems natural, it is not the only possible one. The sum of an infinite series can be determined in other ways. Consider, for example, series (7), which can be written compactly as

As we have already said, its partial sums take the values ​​1 and 0 alternately, and therefore the series does not converge. But if we alternately form pairwise averages of its partial sums (current average), i.e. If we calculate first the average of the first and second partial sums, then the average of the second and third, third and fourth, etc., then each such average will be equal to 1/2, and therefore the limit of pairwise averages will also be equal to 1/2. In this case, the series is said to be summable by the specified method and its sum is equal to 1/2. Many methods of summation have been proposed, which make it possible to attribute sums to fairly large classes of divergent series and thus to use some divergent series in calculations. For most purposes, the method of summation is useful, however, only if, applied to a convergent series, it gives its final sum.

Series with complex terms.

So far, we have tacitly assumed that we are dealing only with real numbers, but all definitions and theorems apply to series with complex numbers (except that the sums that can be obtained by rearranging the terms of conditionally convergent series cannot take arbitrary values).

functional rows.

As we have already noted, the members of an infinite series can be not only numbers, but also functions, for example,

The sum of such a series is also a function whose value at each point is obtained as the limit of the partial sums calculated at that point. On fig. 1 shows graphs of several partial sums and the sum of a series (with x, varying from 0 to 1); s n(x) means the sum of the first n members. The sum of the series is a function equal to 1 at 0 J x x = 1. The functional series can converge for the same values x and disagree with others; in our example, the series converges at –1J x x.

The sum of a functional series can be understood in different ways. In some cases, it is more important to know that partial sums are close (in one sense or another) to some function on the entire interval ( a, b) than to prove the convergence or divergence of the series at individual points. For example, denoting a partial sum n-th order through s n(x), we say that the series converges in the mean square to the sum s(x), if

A series may converge in the mean square even if it does not converge at any single point. There are also other definitions of the convergence of a functional series.

Some functional series are named after the functions that they include. As an example, we can give power series and their sums:

The first of these series converges for all x. The second row converges for | x| r x r x| Ј 1 if r> 0 (except when r is a non-negative integer; in the latter case, the series terminates after a finite number of terms). Formula (17) is called the binomial expansion for an arbitrary degree.

Dirichlet series.

Dirichlet series are functional series of the form S (1/ a n x), where the numbers a n increase indefinitely; An example of a Dirichlet series is the Riemann zeta function

Dirichlet series are often used in number theory.

trigonometric series.

This is the name of functional series containing trigonometric functions; trigonometric series of a special kind used in harmonic analysis are called Fourier series. An example of a Fourier series is the series

F( x), which has the following property: if we take a specific partial sum of series (18), for example, the sum of its first three terms, then the difference between f(x) and this partial sum calculated for some value x, will be small for all values x near 0. In other words, although we cannot achieve a good approximation of the function f(x) at any particular point x, far from zero, taking even very many terms of the series, but for x close to 0, only a few of its terms give a very good approximation. Such rows are called asymptotic. In numerical calculations, asymptotic series are usually more useful than convergent series, since they provide a fairly good approximation with the help of a small number of terms. Asymptotic series are widely used in probability theory and mathematical physics.

Double rows.

Sometimes you have to sum two-dimensional arrays of numbers

We can sum row by row and then add the row sums. Generally speaking, we have no particular reason to prefer rows over columns, but if the summation is done over columns first, the result may be different. For example, consider the double row

Here, each row converges to a sum equal to 0, and the sum of the row sums is therefore also equal to zero. On the other hand, the sum of the members of the first column is 1 and all the other columns is 0, so the sum of the sums over the columns is 1. The only "convenient" convergent double series are the absolutely convergent double series: they can be summed by rows or columns, as well as in any other way, and the amount is always the same. There is no natural definition of the conditional convergence of double series.

What is the sum of all natural numbers? Intuition tells you that the answer is infinity. In mathematical analysis, the sum of natural numbers is a simple example of a divergent series. However, mathematicians and physicists have found it useful to give fractional, negative, and even zero values ​​to the sums of such series. The purpose of my article is the desire to remove the veil of secrecy surrounding the results of the summation of divergent series. In particular, I will use the Sum function (a function for finding partial sums, series, etc. in Mathematica), as well as other functions in the Wolfram Language in order to explain in what sense it is worth considering the following statements:

The importance of labeling formulas with letters A, B, C, and D will soon become clear to you.

Let's start by recalling the concept of a convergent series using the following infinitely decreasing geometric progression.

Common term of the series, starting from n = 0 , is determined by the formula:

Now let's set the sum of the terms of the series from i= 0 up to some final value i = n.

This final amount is called partial sum of a series.

The graph of the values ​​of such partial sums shows that their values ​​approach the number 2 with increasing n:

Applying the Limit function (search for the limit of a sequence or function at a point), we find the limit of the value of the partial sums of this series when n to infinity, which confirms our observations.

The Sum function gives the same result when we sum the terms of a series from 0 to infinity.

We say that a given series (the sum of a given infinitely decreasing geometric progression) converges and what it sum equals 2.

In general, an infinite series converges if the sequence of its partial sums tends to some value as the number of the partial sum increases indefinitely. In this case, the limit value of the partial sums is called the series sum.

An infinite series that does not converge is called divergent. By definition, the sum of a divergent series cannot be found using the partial sum method discussed above. However, mathematicians have developed various ways of assigning finite numerical values ​​to the sums of these series. This amount is called regularized the sum of a divergent series. The process of computing regularized sums is called regularization.

Now we will look at example A from the introduction.

"A" stands for Abel, the famous Norwegian mathematician who proposed one of the regularization techniques for divergent series. In the course of his short life, he died at just 26 years old, Abel achieved impressive results in solving some of the most difficult mathematical problems. In particular, he showed that the solution of an algebraic equation of the fifth degree cannot be found in radicals, thereby putting an end to a problem that had remained unsolved for 250 years before him.

In order to apply the Abel method, we note that the common term of this series has the form:

This can be easily verified by finding the first few values a[n].

As you can see in the graph below, the partial sums of the series take on values ​​equal to 1 or 0, depending on whether the n or odd.

Naturally, the Sum function gives a message that the series diverges.

The Abel regularization can be applied to this series in two steps. First, we construct the corresponding power series.

We then take the limit of this sum at x tending to 1, we note that the corresponding series converges for the values x less than but not equal to 1.

These two steps can be combined, forming, in fact, the definition of the sum of a divergent series over Abel.

We can get the same answer using the Regularization option for the Sum function as follows.

Meaning 1 / 2 seems reasonable, since it is the average of two values, 1 and 0, taken as a partial sum of this series. In addition, the passage to the limit used in this method is intuitive, since when x= 1 the power series coincides with our divergent series. However, Abel was greatly troubled by the lack of rigor that was inherent in calculus at the time, and expressed his concern about it:

“Divergent series are the invention of the devil, and it is a shame to refer to them with any kind of evidence. With their help, one can draw any conclusion he wants, and that is why these series produce so many errors and so many paradoxes. (N. H. Abel in a letter to his former teacher Berndt Holmboy, January 1826)

Let us now turn to example B, which states that:

"B" stands for Borel, a French mathematician who worked in areas such as measure theory and probability theory. In particular, Borel is associated with the so-called “infinite monkey theorem”, which states that if an abstract monkey randomly hits the keyboard of a typewriter for an infinite amount of time, then the probability that he will type some specific text, for example, the entire collected works of William Shakespeare, is different from zero.

In order to apply the Borel method, we note that the common term of this series has the form:

Borel regularization can be applied to rapidly divergent series in two steps. In the first step, we calculate the exponential generating function for the sequence of terms in the given series. The factorial in the denominator ensures the convergence of this series for all values ​​of the parameter t.

Then we perform the Laplace transform of our exponential generating function and look for its value at the point s= 1 .

These steps can be combined, as a result we get, in fact, the definition of the sum of a divergent series over Borel.

We can also use specialized Wolfram Language functions to find the exponential generating function and the Laplace transform:

In this case, the answer can be obtained directly using Sum as follows.

The definition of the Borel sum is reasonable, since it gives the same result as the usual method of partial sums when applied to a convergent series. In this case, we can swap the summation and integration, and then define the Gamma function , in which case we get that the corresponding integral will be equal to 1 and remain simply, in fact, the original sum of the series:

However, in the case of divergent series, it is impossible to swap the signs of the sum and the integral, which leads to interesting results that this regularization method gives.

Borel summation is a universal method for summing divergent series, which is used, say, in quantum field theory. There is a vast collection of literature on the application of Borel summation.

Example C states that:

The “C” stands for Cesaro (his last name is spelled Cesaro in English), an Italian mathematician who made significant contributions to differential geometry, number theory, and mathematical physics. Cesaro was a very prolific mathematician and wrote about 80 papers between 1884 and 1886 before getting his PhD in 1887!

To begin with, we note that the common term of the series, starting from n= 0, has the form:

The graph shows a strong oscillation of the partial sums of this series.

The Cesaro method uses a sequence of arithmetic means of the partial sums of a series in order to suppress oscillations, as shown in the following graph.

Formally speaking, summation by Cesaro is defined as the limit of the sequence of arithmetic means of the partial sums of the series. Calculating this limit for the series from example C, we get the result we expect -1/2 (see graph above).

The Cesaro sum can be obtained directly if we use this type of regularization in the Sum function by specifying the appropriate value of the Regularization option.

The Cesaro summation method plays an important role in the theory of Fourier series, in which series based on trigonometric functions are used to represent periodic functions. The Fourier series for a continuous function may not converge, but the corresponding Ces'aro sum (or Ces'aro mean as it is commonly called) will always converge to the function. This beautiful result is called Fejér's theorem.

Our last example states that the sum of the natural numbers is -1/12.

"D" stands for Dirichlet, a German mathematician who made a huge contribution to number theory and a number of other areas of mathematics. The breadth of Dirichlet's contributions can be judged simply by introducing into Mathematica 10 next code.

Out//TableForm=

Dirichlet regularization gets its name from the concept of “Dirichlet series”, which is defined as follows:

A special case of this series is the Riemann zeta function, which can be defined as follows:

The SumConvergence function tells us that this series converges if the real part of the parameter s will be greater than 1.

However, the Riemann zeta function itself can be defined for other values ​​of the parameter s using the process of analytic continuation known from the theory of functions of a complex variable. For example, when s= -1, we get:

But at s= -1, the series defining the Riemann zeta function is the natural series. From here we get that:

Another way to understand this result is to introduce an infinitesimal parameter ε into our divergent series term, and then find the Maclaurin series expansion of the resulting function using the Series function, as shown below.

The first term in the expansion above tends to infinity as the parameter ε approaches zero, while the third term and all subsequent terms tend to zero. If we discard all terms that depend on ε, then the remaining number -1/12 will just be the Dirichlet sum of the natural series. Thus, the Dirichlet sum is obtained by discarding the infinitesimal and infinitely large terms of the expansion of the series constructed in the way we have described. This is in contradiction with the fact that it is customary to discard only infinitesimal quantities in ordinary mathematical analysis, so the result of summing divergent series according to Dirichlet is not so intuitive.
Stephen Hawking applied this method to the problem of computing Feynman integrals in curved space-time. Hawking's article describes the process of zeta regularization in a very systematic way and it gained a lot of popularity after its publication.

Our knowledge of divergent series is based on the deepest theories developed by some of the best thinkers of the last few centuries. However, I agree with many readers who, like me, feel a bit of a misunderstanding when they see them in modern physical theories. The great Abel was probably right when he called these series "the invention of the devil." It is possible that some future Einstein, with a mind free from all sorts of foundations and authorities, will discard the prevailing scientific beliefs and reformulate fundamental physics so that there will be no room for divergent series in it. But even if such a theory becomes a reality, the divergent series will still provide us with a rich source of mathematical ideas, lighting the way to a deeper understanding of our universe.

Partial sums have no finite limit. For example, rows

diverge.

R. r. began to appear in the works of mathematicians of the 17th and 18th centuries. L. Euler (L. Euler) was the first to come to the conclusion that it is necessary to pose the question, not what is the sum, but how to determine the sum of R. R., and found an approach to solving this issue, close to the modern one. R. r. to con. 19th century did not find application and were almost forgotten. Accumulation to con. 19th century various mathematical facts. analysis again aroused interest in R. r. The question of the possibility of summing series in a certain sense different from the usual one began to be put forward.

EXAMPLES. 1) If you multiply two rows

converging respectively to A and B, then the series obtained as a result of multiplication

may be divergent. However, if the sum of series (1) is defined not as partial sums s n, but as

(2)

then in this sense the series (1) will always converge (i.e., the limit in (2) will exist) and its sum in this sense is equal to C=AB.

2) Fourier series of the function f(x) , continuous at the point x 0 (or having a discontinuity of the 1st kind), may diverge at this point. If the sum of the series is determined by formula (2), then in this sense the Fourier series of such a function will always converge and its sum in this sense is equal to f (x 0) (or, respectively, if x 0 - discontinuity point of the 1st kind).

3) Power series

converges for to the sum and diverges for . If the sum of the series is defined as

(4)

where s n are partial sums of series (3), then in this sense series (3) will converge for all z satisfying the condition Re z

To generalize the concept of the sum of a series in the theory of R. r. consider a certain operation or rule, as a result of which R. r. is placed in a certain, naz. its sum (in this definition). Such a rule is summation, method. So, the rule described in example 1), called. by the summation of arithmetic means (see. Cesaro summation methods). The rule defined in example 2), called. Borel method of summation.

see also Summation of divergent series. Lit.: Voge 1 E., Lecons sur les series divergentes, P., 1928; H and rd and G., Divergent series, trans. from English, M., 1951; R. Cook, Infinite Matrices and Sequence Spaces, trans. from English, M., I960; R e u e r i m h o f f A., Lectures on summability, V., 1969; K n o r r K., Theory and application on infinite series, N. Y., 1971; Z e 1 1 e r K., B e e k m a n n V., Theory der Limitierungsverfahren, B.- Hdlb. - N. Y., 1970. I. I. Volkov.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

See what "Divergent Series" is in other dictionaries:

divergent series- - [A.S. Goldberg. English Russian Energy Dictionary. 2006] Energy topics in general EN divergent series … Technical Translator's Handbook

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