Boundary mark of the raabe with proof. Numerical lavi increased folding. That sufficient mindfulness of a positive-sign numerical series is necessary
In this article, the information is structured, which is necessary for practical application on the topic of number series, depending on the significance of the sumi series to reach the profit.
Review of the article.
Let's take a look at the sign of the positive sign, the sign of the series and the understanding of life. Let's take a look at the standard rows, such as a harmonic row, narrow down the harmonic row, guess a formula for knowing the sum of a geometric progression, which inexorably decreases. After that, let's move on to the power of the rows, which converge, sing on the necessary intelligence of the row and sound sufficient signs of prosperity to the row. The theory is developed by memo solutions of typical applications with report explanations.
Navigation on the side.
The main purpose is to understand.
Let me have a numerical sequence, de 
.
Let's put an example of a numerical sequence: 
.
number series- The sum of the members in the numerical sequence of the mind 
.
As an example of a numerical series, you can put the sum of an infinitely decreasing geometric progression with a standard q = -0.5: 
.
name the leading member of the number series or the k-th member of the series.
For the front butt, the head member of the number series can be seen.
Chastkov's sum of the number series- tse sum mind, de n - deyak natural number. also called the n-th partial sum of the numerical series.
For example, the fourth chastkov sum in a row 
є 
.
Chastkovi sumi 
utvoryuyut neskіchennu sequence of private sums of the numerical series.
For our series, n-a part of the sum is behind the formula for the sum of the first n terms of a geometric progression 
so that mother will come the sequence of private sums: 
.
The number series is called similar yakshcho іsnuє kіntseva between the sequence of chastkovy sums. If between the sequence of partial sums of a numerical series is not іsnuє or not limited, then the series is called rozbіzhnym.
Sum of a similar number series called between the sequence of yogo private sums, tobto, 
.
At our butt, otzhe, a row converge, moreover, yogo suma is more expensive to sixteen thirds: 
.
As a butt of a rozbіzhny row, you can bring the sum of the geometric progression with a larger banner, lower one: 
. n-a chastkovy sum is indicated by viraz 
, and the boundary between private sums is unskinned: 
.
Another butt of a rozbіzhny numerical series is a sum of mind. At tsomu vipadku n-a chastkova sum can be counted as yak. The border between private sums is unskinned 
.
Suma mind 
called harmonious numerical near
.
Suma mind 
de s - deike deisne number, called staggered by the harmonic number row.
It is enough to point the vignette for priming such hardeners, which are often vicorated, it is recommended to remember them.
The harmonic series is divided.
Let's bring the diversity of the harmonic series.
Let's assume that it is low to converge. Todі іsnuє kіntseva between yogo chastkovy sums. With whom you can write down i, which will lead us to equanimity 
.
From the other side, 
Don't call out sumnіvi so nerіvnostі. In such a manner, Nerіvnіst otrimana tells us to those who are jealousy can not be reached, what to say to our allowance about the viability of the harmonic series.
Visnovok: the harmony row to disperse.
SUMA OF GEOMETRIC PROGRESSION IN THE VIEW OF THE SIGNIFICANT q Є CONVERGENT NUMERICAL SERIES, YAKSHO , І LOOKED NEARBY AT .
Let's bring it.
We know that the sum of the first n terms of a geometric progression goes over the formula 
.
When fair 
what to say about the zbіzhnist number series.
For q = 1, there can be a number series 
. Yogo chastkovі sumi perebuvayut yak, and the border between chastkovi sums is unskinned 
, what to say about the rozbіzhnіst are low in different countries.
If q \u003d -1, then the number series will be seen 
. Frequent sums take values for unpaired n and for paired n . From which it is possible to create a non-trivial visnovka, so that between private sums it is impossible and a number of disperse.
When fair 
scho tell about the rozbіzhnist number series.
A VERY HARMONIOUS SERIES CONVERGUE AT s > 1 І IS SEEN AT .
Bringing.
For s \u003d 1, we take away the harmonic series, and we set it apart.
At s the inequality holds for all natural k . Due to the versatility of the harmonic series, it is possible to assert that the sequence of one part sum is unbounded (there are no shards of the final boundary). However, the sequence of partial sums of the numerical series is more unbounded (the skin member of this series is larger than the second member of the harmonic series), also, the harmonization series is aggravated to diverge at s.
Lost to finish the series with s>1.
Let's write down the difference: 
It is obvious that 
Let's write down the unevenness for n = 2, 4, 8, 16, … 
Vykoristovuyuchi tsі results, with a vikhіdnim number you can carry out the following activities: 
Viraz 
є the sum of the geometric progression, the banner of what is good. Oskіlki mi razglyadёmo vpadok at s > 1, then. Tom 
. In such a rank, the sequence of private sums is marked by a harmonic series with s\u003e 1 є growing and at the same time the meanings of the beast, then, there may be a boundary, which indicates the zbіzhnist series. Proof completed.
The number series is called positive sign because all of its members are positive, so 
.
The number series is called sign alternate as signs of yogo sudnіh members of the rіznі. Known number series can be written in 
or 
, de .
The number series is called familiar, as if to revenge the impersonal, both positive and negative members.
The sign-alternating number series є we will call the sign line of the sign series.
Rows 
є sign-positive, sign-ordered and sign-positive.
For the familiar series, it is clear to understand the absolute and mental comfort.
absolutely similar, so that a series of absolute values of its terms converges, then a positive number series converges.
For example, number rows 
і 
converge absolutely, shards converge a series 
, which is the sum of an infinitely slow geometric progression.
The familiar row is called mentally similar like a series to diverge, and a series to converge.
Like a butt of a mentally similar number series, you can induce a series 
. number series 
, Warnings from the absolute values of the members of the output series, which diverge, so that they are harmonious. At the same hour, the exit row is similar, which is easy to install for help. In this order, the number series mentally similar.
The power of similar numerical series.
butt.
Bring the value of the numerical series.
Solution.
Let's write the series in a different way 
. The number series converges, so the harmonic series is made similar for s > 1, and due to the other power of the number series, which converge, the series will converge with the numerical coefficient.
butt.
Chi converge number series.
Solution.
Let's remake the exit row: 
. In this order, we took away the sum of two numerical rows i, moreover, the skins of them converge (marvel at the front butt). Later, by virtue of the third authority of the numerical series, which converge, converge and the external series.
butt.
Bring the value of the number series 
and calculate your sum.
Solution.
The Danish number series can be filed at the sight of two rows: 
Leather from these rows is a sum of infinitely progressive geometric progression, also similar. The third power of the series, which converge, allows you to confirm that the current number series converges. Let's calculate yoga sum.
The first member of the series is one, and the sign of the second geometric progression is 0.5, later, 
.
The first member of the row is є 3, and the banner of the indefinitely progressively progressive geometric progression is 1/3, to that 
.
We quickly take the results for the value of the sum of the output number series: 
The necessary mental health is low.
How the number series converges, between the th k-th term is equal to zero: .
When we have reached the next number of numbers on the zbіzhnіst, we have to follow the next step to change the necessary mind zbіzhnosti. Nevikonannya tsієї umou svіdchit about the rozbіzhnіst numeric series, so, yakscho, then the series diverge.
From the other side, you need to be smart, that your mind is not sufficient. Tobto vikonannya rіvnostі not to talk about the zbіzhnіst number series. For example, for a harmonic row, it is necessary for the mind to win, and the row to diverge.
butt.
Slide the number series to zbіzhnist.
Solution.
Perevirimo necessary for the mental abbreviation of the numerical series: 
Mezha The n-th member of the numerical series is not equal to zero, so the series will diverge.
Sufficient signs of zbіzhnostі sign-positive series.
In case of victorious sufficient signs for reaching the number rows for success, it is gradually brought to converge in such a way that it is recommended to move to the same division in case of difficulty.
That sufficient mindfulness of a positive-sign numerical series is necessary.
For zbіzhnostі positive-sign numerical series 
it is necessary and sufficient, so that the sequence of yogo chastkovy sums of bula is obmezhena.
Let's do it from the sign of the alignment of the rows. Their essence is based on the sequence of the dosledzhuvannogo number series next to each other, zbіzhnіst chi rozbіzhnіst yak vydoma.
The first, the other, and the third signs are equal.
The first sign of the alignment of rows.
Let і - two positive-sign numbers in the series і show unevenness for all k \u003d 1, 2, 3, ... Then, in the zbіzhnosti series, the zbіzhnіst is striking, and in the zbіzhnosti, the zbіzhnіst is evident.
The first sign of porivnyannia vikoristovuetsya more often and more often pressing tool doslіdzhennya numeric rows on zbіzhnіst. The main problem is the pіdbіr vіdpovіdny row for pіvnyannya. A row of equals sounds (ale zavzhdi) is also chosen as the indicator of the step of the th k-th member of the pre-existing indicator of the degree of the number book and the banner of the k-th member of the previous numerical series. For example, let's say, the difference between the indicators of the degree of the numeral and the standard is more 2 - 3 \u003d -1, then, equally, a row is selected from the k-th member, that is, a harmonic series. Let's take a look at the sprat of applications.
butt.
Set zbіzhnіst chi rozbіzhnіst row.
Solution.
Since the boundary of the last member of the row is equal to zero, then the mindfulness of the row is vikonano is necessary.
It doesn't matter if you remember that unevenness is true for all natural ones. We know that the harmony row diverges, therefore, after the first sign of alignment, the outer row is also divided.
butt.
Continue the number series for zbіzhnist.
Solution.
Necessary mindfulness of the numerical series is victorious, shards 
. Obviously vikonannya nervousness 
for whoever natural value k. The series converge, the scales are shaped by the harmonic series є similar to s > 1. In this order, the first sign of the sequence of the rows allows us to state the zbіzhnіst of the output numerical series.
butt.
Vznachte zbіzhnіst chi rozbіzhnіst numeric series.
Solution.
Otzhe, nebhіdnu umovu zbіzhnostі number series vikonano. Which row to choose for the alignment? We ask for a number series, and the choice is assigned to s, respectfully follow the numerical sequence. Members of the numerical sequence grow to infinity. In this order, starting from the last number N (and itself from N = 1619), the members of the sequence will be greater than 2. Starting from the first number N, the nerіvnіst is fair. The number series to descend from the strength of the first power of the rows, to converge, to that to go out of the row, to go, in the first N - 1 member. In such a rank, after the first sign of similarity, there is a series, and due to the first power of the numerical series, which converge, the series will converge.
Another sign of porivnyannya.
Let's go and sign-positive number rows. Yakscho, then zі zbіzhnosti a number of viplivaє zbіzhnіst. Yakshcho, then from the diversity of the numerical series, the diversity is evident.
Consequence.
If so, then from the zbіzhnosti of one row, the zbіzhnіstі іnshoy vyplivaє zbіzhnіst іnshoy, that yakscho zіzbіzhnostі vplіvaє razbіzhnіst.
Doslіdzhuєmo a number of zbіzhnіst s with the help of another signs of equivalence. Like a row, take a row to go. We know the difference between the k-th members of the numerical series: 
In such a rank, after another sign of the equality of the zbіzhnosti of the numerical series, the zbіzhnіst of the output series follows.
butt.
Follow the number series for the zbіzhnіst.
Solution.
Perevirimo necessary for the mentality of the series 
. Umova Vikonan. For zastosuvannya other signs of alignment, we take a harmonic row. We know the difference between the k-members: 
Later, from the diversity of the harmonic row, the diversity of the output row is seen behind another sign of alignment.
For information, we will point the third sign of the alignment of the rows.
The third sign is equal.
Let's go and sign-positive number rows. As from the deyaky number N, the mind wins, then from the zbіzhnosti a row the zbіzhnіst is vibrating, that yakscho from the rozbіzhnosti to the next rozbіzhnіst.
Sign of d'Alembert.
Respect.
The sign of d'Alembert is fair, as if the boundary is not narrow, so 
, then the series converges, so 
, then the series diverge.
Yakshcho, then the sign of d'Alembert gives information about the income and the availability is low and it is necessary to carry out additional work.
butt.
Continue the number series on the zbіzhnist after the d'Alembert sign.
Solution.
Revisiting the necessary mindfulness of the numerical series, between the calculation for:
Umova Vikonan.
Speeding with the sign of d'Alembert: 
In such a rank, it is low to converge.
Radical sign of Kosha.
Come on - a positive sign number series. Yakscho, then the number series converge, yakscho, then the series diverge.
Respect.
The radical sign of Kosh is fair, as if the boundary is not skinny, so 
, then the series converges, so 
, then the series diverge.
As a radical sign of Kosh, it does not give information about the income, but the difference in a number of ways and the need for additional follow-up.
Sound to easily look at the vipades, if it is better to vikoristovuvat the radical sign of Koshі. Characteristic є vipadok, if the zagalny member of the number series є ostentatious static viraz. Let's take a look at the sprat of applications.
butt.
Follow the positive number series for zbіzhnist for additional radical signs of Koshі.
Solution.
. Behind the radical sign of Kosh is taken 
.
Otzhe, low converge.
butt.
Chi converge number series 
.
Solution.
Speeding by the radical sign of Kosha 
Otzhe, the number series converge.
Integral sign of Cauchy.
Come on - a positive sign number series. We add a function of a non-permanent argument y = f (x), similar to a function. Let the function y = f(x) be positive, uninterrupted and falling on the interval , de ). Todi at the same time non-consecutive integral converge doslidzhuvany number series. Yakshcho unclassified integral disperse, then the outer row can also disperse.
When reversing the change of the function y = f(x) on the interval, you may need a theory from the division.
butt.
Complete the number series with positive members for the feasibility.
Solution.
Necessary mindfulness in a row of vikonan, oskelki 
. Let's look at the function. It is positive, uninterrupted and falling at intervals. The uninterruptedness and positivity of this function does not call for doubt, but on the decline we hear a little more. Let's get lost: 
. It is negative for the interval, therefore, the function changes for the same interval.
In vipads, if the signs of d'Alembert and Cauchy do not give a result, sometimes positive signs can give signs based on equal parts with lower rows, which converge or diverge more "more", lower row of geometric progression.
We will induce, without proof, the formulation of some cumbersome signs of the profitability of the rows. The proofs of these are also based on the ordering theorems 1–3 (Theorem 2.2 and 2.3) of a given series of decimal series, the feasibility of which has already been established. Qi proof can be known, for example, from the fundamental assistant G. M. Fikhtengolts (vol. 2).
Theorem 2.6. Sign of Raabe. As for members of a positive number series, starting from the decimal number M, unevenness is calculated
(Rn £ 1), "n ³ M, (2.10)
then the series converges (diverges).
Sign of Raabe at the boundary form. As for the members of the appointed higher, the mind is consecrated in a row
Note 6. If the signs of d'Alembert and Raabe are equal, it can be shown that the other is significantly stronger than the first.
Yakshcho for a number of basic boundaries
then for the sequence of Raabe, there is a boundary
In this order, just as the sign of d'Alembert gives evidence for food about zbіzhnіst or razbіzhnіst in a row, then the sign of Raabe is also given, moreover, the number of vipadki is only two of the possible values of R: + і -. All other variations of the final R 1, if the sign of Raabe gives a solid evidence for food about the diversity of the row, give good luck D \u003d 1, then it’s good, if the sign of D’Alembert does not give a solid evidence for food about the diversity of the row.
Theorem 2.7. Kummer sign. Come on (cn) - enough sequence of positive numbers. As for members of a positive number series, starting from the decimal number M, unevenness is calculated
(Qn £ 0), "n ³ M, (2.11)
then the series converge 
.
Kummer sign at the boundary form. Yakshcho for a designated vishche in a row is a clear boundary
then the series converge .
From the signs of Kummer, as a legacy, it is easy to take away the sign of d'Alembert, Raabe and the signs of Bertrand. Stop going out, like a sequence (сn) take
cn=nln n, "n н N,
for which row
disperse (the variability of which row will be shown in the butts of this paragraph).
Theorem 2.8. Bertrand's sign at the boundary form. Like for members of a positive number series the sequence of Bertrand
(2.12)
(Rn - succession of Raabe) may be between
then the series converges (diverges).
Below we formulate the sign of Gaus - the most strained sign of the succession of the rows of d'Alembert, Raabe and Bertrand. The Gaussian sign encloses the entire frontal sign and allows you to fold meaningfully folding rows, but, on the other side, for the first stop, it is necessary to carry out more subtle adjustments in order to take asymptotic expansion of the lower order of the next order of magnitude.
Theorem 2.9. Gaus sign. Likewise, for members of a positive number series, starting from the decimal number M, equality is calculated
, "n ³ M, (2.13)
de l and p are constant, and tn is a delimited value.
a) for l> 1 or l = 1 і р> 1 the series converges;
b) for l< 1 или l = 1 и р £ 1 ряд расходится.
2.5. Integral sign Cauchy-Maclaurin,
“telescopic” sign of Kosh and sign of Ermakov
Seen more signs of success in a series of foundations on the theorems of equalization and є sufficient, so that signs this row if you can make a sing-song about yoga behavior, but if you remember the signs for a new one not vikonan, then you can’t make anything about stability in a row, you can converge like this and diverge.
INTEGRAL INTERY INTENTION OF THE MALORENA VID VID VIZHINIKH VISHISH behind the snake, being the UNITY IN-HEART, and such, BAZISISE ON INTERNICAL SUMI (RAS) INTORYALARYARIVA, I DEMARETICARY RARIMALIVAYAVAYAVAYAVAYAVAYAVAYAVAYAVARYARYARIVARYAVAYAVARYAVAYAVARYAVAYARYARIVARYARYARYARIVARYARYARYARYARYARYARYARY. Tsey vzaimozv'yazok easily prostzhuєtsya on the butt and sign of porіvnyannya, analogues of which may be a place for obscure іntegraіv that yogo formulas may be literally zbіgayutsya z formularyovannya for rows. A similar analogy is also observed in the formulas of sufficient signs of the sufficiency of sufficient numerical rows, as they will be in the offensive paragraph, that sign of the sufficiency of unsighted integrals - such as the signs of the sufficiency of Abel and Dirichlet.
Below will be placed also the “telescopic” sign of Kosh and the original sign of the rows of rows, taken away by the Russian mathematician V.P. Yermakovim; Ermakov’s sign for its tightness can be approximately that area of zastosuvannya, like the Cauchy-Maclaurin integral sign, to protest against the formulary of terms and understand the integral calculation.
Theorem 2.10. Sign of Cauchy-Maclaurin. Go for the members of a positive number series, starting from the current number M, equanimity wins
de function f(x) is not visible and does not grow on the direct line (x ³ M). The number series converges more or less the same, if the ambiguous integral converges
So the series converge, as if there is a boundary
, (2.15)
that series diverge, as if between I = + ¥.
Bringing. Through respect 3 (div. § 1) it is obvious that without an exchange of cohesion, we can enter M = 1, shards, having added (M - 1) members in a row and having added a change k = (n - M + 1), we come to consider the row, for which
, 
,
i, obviously, before looking at the integral.
It is worthy of note that the function f(x), which is not growing on a straight line (x ³ 1), satisfies the minds of integration according to Riemann on any end-of-the-day interval, and the view of a similar non-linear integral may be sensible.
Let's move on to confirmation. On any segment of a single period m £ x £ m + 1, through the non-expansion f(x) we can see the unevenness
Integrating yoga according to the vіdrіzku and speeding up in the form of power sing integral, take away the unevenness
, . (2.16)
Subsumuovuyuchi tsі nerіvnosti term-by-term vіd m = 1 to m = n, we take
Oskіlki f(х) is not a function, then the integral
є uninterrupted uninterrupted function to the argument of A. Todi
, .
Zvіdsi i z nerіnostі (15) vyplivaє, scho:
1) Yakscho I< +¥
(т. е. несобственный интеграл сходится), то и неубывающая
последовательность частичных сумм 
fenced, then the row converge;
2) so that I = + ¥ (so that the inconsistent integral diverges),
those indestructible succession of private sums is not limited, that is, a number of disperse.
From the other side, knowing, from the nervousness (16) is obsessed:
1) yakscho S< +¥
(т. е. ряд сходится), то для неубывающей uninterrupted function I(A), "A ³ 1 if the number n is such that n + 1 ³ A, i I(A) £ I(n + 1) £ Sn £ S, and then, 
, i.e., the integral converges;
2) if S = +¥ (so the row diverges), then for whatever it is to achieve the great A іsnuє n £ A such that I (A) ³ I (n) ³ Sn - f (1) ® +¥ (n ® ¥), so that the integral diverges. What did it take to bring.
We will induce, without proof, two signs of prosperity.
Theorem 2.11. "Telescopic" sign Kosha. A positive number series, the members of which change monotonously, converge more or less the same, if the series converges.
Theorem 2.12. Yermakov's badge. Let the members of the positive number series be such that from the deyakogo number M0, equanimity
an = (n), "n ³ M0,
de function (x) is shmatkovo-perpetual, positive and changes monotonously at x ³ M0.
The same way the number M ³ M0 is the same, so that for all x ³ M the unevenness is
,
then the series converges (diverges).
2.6. Apply a sign of prosperity
Following the help of Theorem 2, it is easy to follow the succession of the next series
(a > 0, b ³ 0; "a, b Î R).
If it is 1, then the necessary sign of prosperity (dominion 2) will be destroyed (div. § 1).
,
Otzhe, a number of disperse.
If a > 1, then сn may be a small estimate, because of which the geometric progression of the analyzed series is low.
converge to the signs of alignment 1 (theorem 2.2), the shards may be uneven
,
and the series descend like a series of geometric progression.
We show the diversity of a number of rows, which show signs of alignment 2 (last 1 of Theorem 2.2). Row
disperse, shards
.
disperse, shards
.
disperse, shards
.
(p > 0)
disperse, shards
.
go for the d'Alembert sign (Theorem 2.4). Deisno
.
converge for the sign of d'Alembert. Deisno
.
.
converge after the Cauchy sign (Theorem 2.5). Deisno
.
Let's aim the butt of zastosuvannya signs of Raabe. Let's look at the row
,
design (k)!! means the addition of all paired (unpaired) numbers from 2 to k (from 1 to k), so k is paired (unpaired). Vikoristovuyuchi sign d'Alembert, otrimaemo
In this rank, the sign of d'Alembert does not allow the growth of singing affirmation about the low income. We must sign Raabe:
otzhe, the series converge.
Let us apply the Cauchy–Maclaurin integral sign.
Harmonious Row
converge and diverge at the same time with an undefined integral
It is obvious that I< +¥ при p >1 (integral converge) і I = + ¥ for p £ 1 (diverge). In this way, the output series also converges for p > 1 and diverges for p £ 1.
diverge at the same time with an inconsistent integral
in such a rank, the integral diverges.
|
|
§ 3. Significant number series
3.1. Absolutely that mindfulness of the rows
In this paragraph, there is an emphasis on the power of the rows, the members of which are speech numbers with a sufficient sign.
Vision 1. Number series
is called absolutely similar, so that the series converges
Appointment 2. A number series (3.1) is called such that it converges mentally or non-absolutely, so that series (3.1) converges and series (3.2) diverges.
Theorem 3.1. If a series converges absolutely, vin converges.
Bringing. Consistent up to the Cauchy criterion (Theorem 1.1), the absolute efficiency of the series (3.1) is equivalent to
" e > 0, $ M > 0 so that " n > M, " p ³ 1 Þ
(3.3)
Since the modulus of the sum of a number of numbers does not exceed the sum of the sum of their modules (“irregularity of the tricot”), then from (3.3) the unevenness is evident (valid for the same, as in (3.3), numbers e, M, n, p)
Loss of the rest of the unevenness means the defeat of the minds by the criterion of Kosh's series (3.1), hence, this series converges.
Conclusion 1. Let series (3.1) converge absolutely. We add up the positive terms of the series (3.1), renumbering them in order (as the stench grows in the process of increasing the index), a positive number series
, (uk =). (3.4)
Similarly, we add up the modules of the negative terms in the series (3.1), renumbering them in order, the next positive number series:
, (Vm = ). (3.5)
So the series (3.3) and (3.4) converge.
If we take the sum of the series (3.1), (3.3), (3.4) up to the letters A, U, V, then the following formula is true
A = U - V. (3.6)
Bringing. The sum of the series (3.2) is significant in terms of A*. By Theorem 2.1, it is possible that all the partial sums of the series (3.2) are exchanged by the number A*, and the partial sums of the series (3.4) and (3.5) appear in the result of subsuming a part of the terms of the partial sums of the series (3.2), then it is obvious that they stink more exchanged by the number A *. Todi, introducing new signs, obsessed with nervousness
;
From these, through Theorem 2.1, we can see the infinity of the series (3.4) and (3.5).
(3.7)
If the numbers k and m lie in p, then it is obvious that when p ® ¥ one hour k ® ¥ and m ® ¥. Then, passing in equality (3.7) to the boundary (all the boundaries are based on Theorem 3.1 and beyond), it is acceptable
i.e., equality (3.6) has been proved.
Conclusion 2. Let series (3.1) converge mentally. If the series (3.4) and (3.5) diverge, then the formula (3.6) for series that converge intelligently is not correct.
Bringing. If we look at the n-th part sum of the series (3.1), then, as we can prove in advance, її can be written
(3.8)
On the other hand, for the n-ї partial sum of the series (3.2), one can write similarly viraz
(3.9)
It is acceptable not to accept, i.e. let one of the series (3.3) or (3.4) converge. Then the formulas (3.8) through the abbreviation of the series (3.1) show that the other series (likely (3.5) or (3.4)) converge like the difference of two series, which converge. And even though formulas (3.9) show the efficiency of the series (3.2), then the absolute efficiency of the series (3.1) supersedes the intellect theorems about the intellectual efficiency.
In such a rank (3.8) and (3.9) weeping, that so
what it was necessary to bring.
Respect 1. Happy power for the ranks. The sum of an inexhaustible series of things is considered in the sum of the final number of elements, which includes the boundary transition. Therefore, the main characteristics of kіntsevih sums are often ruined for rows, or they are saved for the sake of singing minds.
So, for final sums, there may be a place for controversy (associative) law, but for yourself: the sum does not change, as the elements of sumi grouping in any order
Let's take a closer look at the grouping (without rearrangement) of the members of the numerical series (3.1). Significantly growing sequence of numbers
and introduced value
The same row, subtracted in a more significant way, can be written in
In the guidance below, without proving the theorem, a few of the most important strongholds were chosen, connected with the successive power of the ranks.
Theorem 3.2.
1. If a series (3.1) converges and has the sum A (to achieve mental efficiency), then a sufficient series of the form (3.10) converges and may have the same sum A. For a series that converges, may be dominated.
2. In the case of such a series of species (3.10), there is no efficiency in the series (3.1).
3. If the series (3.10) is taken away by special groups, so that in the middle of the skin of the arms there are warehouses of less than one sign, then the efficiency of the series (3.10) is the same for the series (3.1).
4. If series (3.1) is positive and if any series (3.10) can converge, then series (3.1) converges.
5. Since the sequence of members in the series (3.1) is infinitely small (tobto an) and the number of additions in the skin group - the members of the series (3.10) - is surrounded by one constant M (tobto nk –nk–1 £ M, "k = 1, 2, ... ), then the efficiency of the series (3.10) shows the efficiency of the series (3.1).
6. If the series (3.1) converges intelligently, then without permutation of the rows it is possible to group the members of the series in such a way that the subtraction series (3.10) will be absolutely similar.
2. Change of power for the rows. For final numerical sums, there may be a shifting (commutative) law, but for itself: the sum does not change with any permutation of dodankiv
de (k1, k2, …, kn) is a sufficient permutation from the set of natural numbers (1, 2,…, n).
It appears that similar power can be placed for rows that absolutely converge, and do not win for mentally similar rows.
Let it be mutually unambiguous in the multiplication of natural numbers on itself: N ® N, so that the skin natural number k is given a single natural number pk, moreover, the impersonal number creates the entire natural series of numbers without overshoots. Significantly, the series is subtracted from the series (3.1) for an additional additional permutation, which in turn confirms the designated more fermentation, in such a rank:
Rules for the stagnation of shifting power in a series of changes in inductions below without proof in Theorems 3.3 and 3.4.
Theorem 3.3. If the series (3.1) converges absolutely, then the series (3.11), subtracted by a sufficient permutation of the terms of the series (3.1), also converges absolutely in the same amount as the other series.
Theorem 3.4. Riemann's theorem. If the series (3.1) converges intelligently, then the segmented series can be rearranged in such a way that the total amount is more equal to the given number D (finishing or not reduced: ±¥), otherwise it will not be assigned.
On the basis of Theorems 3.3 and 3.4, it is easy to establish that the intellectual profitability of a series of occurrences is due to mutual redemption growth station n-th private sum at n ® ¥ for the account of adding up to the sum of either positive or negative additions, and it is smart for a row to lie in the order of members of a row. The absolute value of the series is the result of a swedish change in the absolute values of the members of the series
and do not lie in the order of their straightening.
3.2. Cherry row sign. Leibniz sign
In the middle of the familiar rows, there is an important private class of rows - rows that are marked.
Designation 3. Come on - sequence of positive numbers bп > 0, "n N N.
is called a next-to-next sign. For series in the form (3.12) there may be such a hardening.
Theorem 5. Leibniz sign. As a sequence, it is composed of the absolute values of the members of the series (3.8), which is drawn, monotonously changes to zero
bn > bn+1, "n н N; (3.13)
then such a sign alternating series (3.12) is called the Leibniz order. The Leibniz series must converge. For a surplus next to Leibniz
maє mistse otsіnka
rn = (–1) nqnbn+1, (0 £ qn £ 1) "nнN. (3.14)
Bringing. Let's write down the partial sum of the series (3.12) with the guy's number of additions in the view
Behind the mind (3.13) the skin of the shackle at the right part of the virase date, Otzhe, zі zrostannyam k sledovnіst monotonously zrostaє. From the other side, whether any member of the B2k sequence can be recorded at a glance
B2k = b1 – (b2 – b3) – (b4 – b5) –… – (b2k–2 – b2k–1) – b2k,
and the shards behind the mind (3.13) have a positive number in the skin of the arms of the remaining evenness, then, obviously, the unevenness
B2k< b1, "k ³ 1.
In such a rank, perhaps monotonously growing and intermingled with the beast, the sequence, and such a sequence for the leading theorem from the theory between
B2k-1 = B2k + b2k,
and vrahovyuchi, that the main term of the series (according to the mind’s theorem) is equal to zero when n ® ¥, it is necessary
In this manner, it was brought to light that the series (3.12) for mind (3.13) converge that yoga sum is more expensive Art.
We bring the estimate (3.14). It is shown more clearly that the partial sum of the pairwise order B2k, monotonously increasing, jump between Y - the sum of the row.
Let's look at the chastkovі sumi unpaired order
B2k–1 = b1 – (b2 – b3) – (b4 – b5) – … – (b2k–2 – b2k–1).
From this point of view, it is obvious (shards of vikonano umova (3.13)), that the sequence changes and, therefore, according to the more advanced of its borders, the beast. In this manner, brought nerіvnіst
0 < B2k < B < B2k–1 < b1. (3.15)
Now look at the excess row (3.12)
as a new sign is drawn by the series with the first member bp + 1, then for which series on the basis of the unevenness (3.15) can be written with paired and unpaired indices
r2k = b2k+1 – b2k+2 + …, 0< r2k < b2k+1,
r2k–1 = – b2k + b2k+1 – …, r2k< 0, | r2k–1 | < b2k.
Also, it has been shown that the excess of the Leibniz series has the sign of its first term i less than the absolute value, i.e. the value (3.14) is counted for it. The theorem has been completed.
3.3. Signs of prosperity of prevіlnyh numerical rows
In this subparagraph, without proof, sufficient signs of adventitiousness are introduced for numerical rows with members, є sufficient real numbers (be it a sign), add tsі signs of adventitiousness in rows with complex members.
2) sequence - similar to zero (bp ® 0 at n ® ¥) sequence with a fringed snake.
Then the series (3.16) converges.
Theorem 3.9. Dirikhle sign. Let the members of the numerical series (3.16) please the minds:
the sequence of private sums is low (irregularities (3.17));
2) the sequence is a monotonous sequence that goes to zero (bp ® 0 when n ®¥).
Then the series (3.16) converges.
Theorem 3.10. The other is marked with the sign of Abel. Let the members of the numerical series (3.16) please the minds:
1) the series converge;
2) succession - a fair succession with a fringed snake.
Then the series (3.16) converges.
Theorem 3.11. Abel sign. Let the members of the numerical series (3.16) please the minds:
1) the series converge;
2) succession - monotonous obmezhenoyu succession.
Then the series (3.16) converges.
Theorem 3.12. Cauchy's theorem. If the rows and converge absolutely the same їх sums are equal in the same way A and B, then the series of foldings from the existing creations in the form aibj (i = 1,2,…, ¥; j = 1,2,…,¥), numbering y be it order, so converge absolutely that yogo sum dorivnyu AB.
3.4. Apply
Let's take a look at the back of the sprat and apply the absolute efficiency of the rows. Lower vvazhaemo, scho change x can be a real number.
2) diverge for |x| > f tієї w sign of d'Alembert;
3) diverge after |x| \u003d e after the sign of d'Alembert in non-existent forms, shards
due to the fact that the exponential sequence, which stands at the banner, pragne to its boundary, monotonously growing,
(a ¹ 0 - actual number)
1) converge absolutely for | x / a |< 1, т. е. при |x| < |a|, так как в to this particular type may be a series of foldings from the terms of the spacing geometric progression with the standard sign q = x/a, or behind the radical Cauchy sign (Theorem 2.5);
2) diverge after |x/a| ³ 1, then for | x | ³ | a |
|
Formulation at the boundary form
Respect. Yakscho It is impossible to open the virus (winning file texvc NOT knowledge; Math/README - proof of the alignment.): R=1, then the sign of Raabe does not give any information about the success of the series.
Bringing
The proof is grounded on zastosuvanni zagalnennoy signs of porіvnyannya when porіvnyannі z zagalnenim harmonіy next
Div. also
- The badge of d'Alembert's wealth is a similar badge, based on suicidal members.
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Literature
- Arkhipov, R. I., Sadovnichy, St A., Chubarikov, St N. Lectures mathematical analysis: Assistant to the university and ped. vuziv / For red. V. A. Sadovnichiy. – M.: Vishcha shkola, 1999. – 695 p. - ISBN 5-06-003596-4..
- - article from the Mathematical Encyclopedia
Posilannya
- Weisstein, Eric W.(English) on the Wolfram MathWorld website.
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Taken in Kummer's meaning as a disparate series (12.1) harmonic series
I can't help it
Otriman of the sign of profitability can be formulated in such a way.
Theorem (sign of Raabe's abbreviation). Row
converge
Tsey row to disperse, as if, pochinayuchi from deyakogo bude
The boundary form of the Raabe sign looks like this:
then the series (12.9) converges, but
want to part.
Raabe’s sign of comfort is signifi- cantly sensitive, the lower one is similar to d’Alembert’s sign of comfort. Actually, there, de sign of d'Alembert, taken from the yogo boundary form, setting the order (12.9):
there the sign of Raabe is given.
Similarly for the row, for the rozbіzhnіst of any indication of the d'Alembert sign, after the sign Raabe will be
1. Let's look at the row
Here so scho with skin specific x
that zastosuvannya signs of d'Alembert here to no avail. Oznaka w Raabe give
It can be seen that the row converges when viewed, but when it diverges. It is commendable that for a row (12.10) it transforms into a harmonious one, which, as you know, disperses. Those that the sign of Raabe in its own (non-existent) form establishes the disparity of the harmonic series, cannot be respected by an independent result, since the warehouse sign of Raabe itself is firmly rooted in this disparity.
Warehouse for the replacement of the suicidal members of the row:
Allocation right-handed In tsomu and offensive butts, we will be corystuvatisya with boundary signs of zbіzhnostі. Tse means that we will not be able to increase the value of the change. To that skin, the coming steps will be with the increase of the inexorably small good order against the front. Letting go of all the steps, starting from the day, we will work for a pardon, as if it were not only absolutely small, but equally with the rest of the dying members. Tsya vodnosna pardon will be less, the more significant and familiar at the border with unfenced growth. Depending on the necessary accuracy, it is necessary to minimize the Taylor formulas for similar functions with the same number of terms. Dali mi we will use the sign of the word, as if they are one and the same in magnitude, small equals with the same accuracy, as to give the omission of the inscribed limbs.
On the back of the head we are surrounded by members of the logarithms and the root, which is no more revenge at the step for pershu. We will be mother
Otzhe, and the sign of d'Alembert's prosperity cannot be given to us here.
Enthusiasm...