Follow the vicarious row, vicorist to the sign of d'Alembert. Numerical rows: appointment, power, signs of wealth, butt, decision. Sufficient signs of life in a positive sign series

Jean Léron d'Alembert was a famous French mathematician of the 18th century. Vzagali D'Alembert was specializing in differential equalizations and on the basis of his achievements, he was engaged in ballistics, so that Yogo Majesty could fly harmonic nuclei better. At the same time, without forgetting about the numbers, it was not without reason that the ranks of the Napoleonic troops converged and parted so clearly.

Before that, how to formulate the sign itself, a look at an important food:
If it is necessary to zastosovuvat sign of d'Alembert's wealth?

We'll start from repetition. Let's guess the ups and downs, if it's necessary to stop the finds boundary sign. Boundary sign porіvnyannya zastosovuєtsya only, if the main member of the series:
1) The bannerman has a rich member.
2) Rich terms are found in the number book, and in the banner.
3) One or the other richly divided can be rooted.

The main rethinking signs of d'Alembert are as follows:

1) Before the final member of the series (“stuffing” of the series), enter the same number in the steps, for example,, and so on. Moreover, it’s not important, it’s important, where the contraption is roztashovuetsya, it’s important to the numeral officer or the bannerman, it’s important that it’s there.

2) Factorial enters before the start member of the series. What is factorial? Nothing collapsible, factorial - it's just a burning record to the creator:

…

…

! When vikoristanny signs of d'Alembert, we happen to write the factorial report. Like i in the front paragraph, the factorial can be spread in the mountains or below the fraction.

3) Like in the zagalny member of the series є “languide of multipliers”, for example,. This vapadok trapleyaetsya rarely, ale! When such a series is completed, pardons are often allowed - divas. Example 6.

Together with the steps of chi (i) factorials in the beginning of the series, rich terms often grow, but it doesn’t change right - it’s necessary to win the sign of d’Alembert.

In addition, the primal member of the series may have one hour at a time and a step and a factorial; you can use two factorials, two steps, it’s important that there was I want to from the examined points - and ce reconsidered signs of d'Alembert.

Sign of d'Alembert: Look at positive number series. How is it between the extension of the offensive member to the anterior one: then:
a) At a row converge
b) At a row disperse
at Pri the sign is not given. It is necessary to win another sign. It’s more common to go alone at that vapadka, if the signs of d’Alembert are trying to get stuck there, it’s necessary to beat the boundary sign of equalization.

Who has problems between the borders, or unreasonable between, go back to those Mezhі. Apply solution. Without rozuminnya between that vminnya rozkrivat invisibility far away, unfortunately, do not stick out. And now apply the dovgoochіkuvani.

butt 1
Mi, scho in the sleeping member is low, but it is correct to reconsider those who need to win the sign of d'Alembert. On the other hand, the decision is made, the commentary is below.

Victory to the sign of d'Alembert:

converge.

(1) We add the position of the offensive member to the anterior one: . Mind bachimo, scho is a hot member of the series. In order to take away the next member of the row, it is necessary Deputy Submit: .
(2) Let's take some extra surface shot. For the singing of the decision, you can skip this part.
(3) The arms of the dialer are curved. At the bannerman, we blame the fourth step.
(4) Fast on . We blame the constant for the boundary. At the numeral in the bows, similar dodanki are induced.
(5) Insignificance is established in the standard way - by subdividing the numeral and banner into "en" at the senior level.
(6) Divide the numerals into banners one by one, and it is possible to make additions, so as to overcome zero.
(7) Let's just follow and work on the sign, that with a whisker about those who follow the sign of d'Alembert, a series of converge.

In the open butt, in the open member, the polynomial of the 2nd degree is low. What work, how is there a rich member of the 3rd, 4th or higher level? On the right, in the fact that the rich term of the higher level is given, then it is difficult to blame the opening of the bow. In this way, you can zastosovuvat "turbo"-method of solution.

butt 2 Let's take a similar series and do it for profit
Back to top solution, then comments:

Victory to the sign of d'Alembert:

In such a rank, doslidzhuvany row converge.

(1) We make a deposit.
(2) Let's take some extra surface shot.
(3) We look at the viraz at the numeralist, that viraz at the bannerman. My bachimo, that in the number book it is necessary to open the arches and lead to the fourth step: what do you want to work with? In addition, for those who do not know Newton's binomial, it may seem unimaginable. Let's analyze the senior step: if we open the arches in the mountains, then we take away the senior step. Below we have such the most senior step:. By analogy with the front butt, it is obvious that we have a mezhіde loneliness with the punctuated rozpodіlі of the numeral and banner. Abo, as mathematicians seem, richly articulated - one order of growth. In such a rank, you can circle the setting with a simple olive and immediately say that this thing is not alone. Similarly, we deal with another pair of rich members: and stink also one order of growth, that yoga setting is pragne alone.

Indeed, such a “hack-work” could be done in Example No. 1, but for a rich member of the 2nd stage, such a decision still looks undignified. Especially I work like this: if it’s a rich member (or a rich member) of the first or another stage, I’ll use the vicorist “longer” method of perfecting Butt 1. Likewise, the rich member of the 3rd and higher levels will traverse, I’ll use the vicorist “turbo” method after the second butt 2.

butt 3 .

Look at typical stocks with factorials:

butt 4 Continue the row to the left

Before the zagalny member of the series, enter і steps, і factorial. It dawned on me, like daylight, that the sign of d'Alembert needed to be victorious here. We see it.

In such a rank, doslidzhuvany row disperse.

(1) We make a deposit. Let's repeat it one more time. For umovoyu zagalny member are low: . To take away the offensive member of the row, deputies need to provide, in this order: .
(2) Let's take some extra surface shot.
(3) Vіdshpuєmo simku vіd step. Factorials are written in a report. Yak tse zrobiti - div. the beginning of the lesson.
(4) Everything that can be shortened is fleeting.
(5) The constant is blamed for the boundary sign. At the number book, the arches are crooked.
(6) Non-significance is assessed in the standard way - by subdividing the numeral and banner into “en” at the senior level.

butt 5 Continue the row to the bottom. More solution below.

butt 6 Continue the row to the left

Rows of people grow up in rows, like at their own beginning to avenge lances of multipliers, whose type has not yet been seen in a row. How to complete a series of multipliers with a "lansy"? Victory to sign d'Alembert. Ale, back to back, for the sake of understanding what is being said, we will write down a series of reports:

From the layout of mi bachimo, that at the skin offensive member of the row, an additional multiplier is added to the bannerman, that is, like a burning member of the row, then the offensive member of the row:
. The axis here is often automatically pardoned, formally recording after the algorithm what

A clear eye of decision can look like this: Victory to the sign of d'Alembert:
In such a rank, doslidzhuvany row converge.
RADICAL SIGN OF KOSH

Augustin Louis Cauchy is a famous French mathematician. You can tell Kosh's biography whether you are a student of a technical specialty. At the hired farbakhs. It’s not a bad name that is hung on the first version of the Eiffel Tower.

The sign of zbіzhnosti Koshі for positive numerical rows, in addition to the surprisingly well-looked-for sign of d'Alembert.

Radical sign of Koshі: Look at positive number series. Yakscho є boundary: then:
a) At a row converge. Zokrema, the series converge for .
b) At a row disperse. Zokrema, a row to disperse for.
at Pri the sign is not given. It is necessary to win another sign. It means that if the sign of Cauchy does not give us advice on food about the economy in a row, then the sign of d'Alembert will not give us advice. But if the sign of d'Alembert is given, then the sign of Kosh can be "repaired" as a whole. Tobto is a sign of Koshі є in tsomu sensі a strong sign.

When is it necessary to vicorate the radical sign of Kosh? Radical signs of Kosh sound vikoristovu in quiet vipadkah, if a hot member is in a row POVNISTTYU to be in the step lie down like "en". Abo if the root of "good" rises from the hard member is low. There are more exotic vipads, but they don’t clog your head with them.

butt 7 Continue the row to the left

Mi Bachimo, that the main member of the series is constantly changing under the step, which is to fall in, then, it is necessary to win the radical sign of Kosh:

In such a rank, doslidzhuvany row disperse.

(1) We make out the main member of the series of sub roots.
(2) We rewrite the same ones, only already without a root, victorious power of steps.
(3) At the showman, we divide the number by term for the banner, indicating that
(4) As a result, we have become insignificant. Here you can drink in the old way: ring in the cube, ring in the cube, then divide the numeral and banner on en at the senior level. ale in to this particular type Efficient solution: you can add the numeral and the banner by term under the step-constant. For usunennya non-significance, we divide the number and banner on (the senior step).
5
(6) It is brought to a path, we note that we are robimo visnovok about those who are in a row to disperse.

And the axis is a simple butt for an independent vision:

butt 8 Continue the row to the left

І more sprat of typical applications.

On the outside, the solution is the one below.

butt 9 Continue the row to the left
Vikoristovuemo radical sign Kosha:

In such a rank, doslidzhuvany row converge.

(1) We place the head member in the row of root.
(2) We rewrite the same, but already without a root, with which we open the arches, the vicorist formula of the short multiplication:.
(3) At the pokanik, we divide the number by term for the znamennik, and it is indicated that.
(4) The insignificance of the mind is taken away. Here you can add the number to the banner on the “en” at the senior level at the arch. It was similar to us when we were talking about the hour of the wedding another miraculous boundary. But here the situation is different. Yakby coefficients at higher steps of the boule the same, for example: , then the trick from the joint roses was no longer passed, and it would be necessary to win another miraculous boundary. Ale, we have q coefficients different(5 and 6), it is possible (and necessary) to divide by term (before speech, navpaki - a friend of the miraculous boundary with rіznih coefficients at senior levels no longer continue).
5
(6) Insignificance has been usuned, the simplest boundary has been lost: Why in infinitely great step pragna to zero? Because the basis of the step is satisfied with the nervousness. If anyone has doubts about justice between me, then I don’t hesitate, I’ll take a calculator to hand:
Yakscho something
Yakscho something
Yakscho something
Yakscho something
Yakscho something
… and so on. to inexcusability - tobto, between:
(7) It is possible, how and robimo visnovok, to the one that the series converges.

butt 10 Continue the row to the left

This is an example of an independent solution.

Occasionally, a provocative butt is shown on the cherry, for example:. Here at the showman there is a step nі "en" only a constant. Here it is necessary to square the numeral and the banner (to see rich terms), and then to trim the algorithm of the statistics Rows for teapots. In a similar butt, it may be necessary either to have a low sign of profitability or a boundary sign of equalization.
INTEGRAL SIGN OF KOSH

I rejoice quietly, who has rottenly acquired the material of the first course. In order to establish the integral sign of Kosh, it is necessary to have a greater or lesser amount of knowledge about the difference, the integration, and also the mother of the calculation non-consecutive integral first kind. At the assistants mathematical analysis The integral sign of Cauchy is given mathematically suvoro, formulating the sign even more primitively, albeit apparently. I apply once for clarification.

Integral sign of Koshі: Look at positive number series. Tsey series converge and diverge

butt 11 Continue the row to the left

Mayzhe classic. Natural logarithm and as a bad thing.

The main rethinking of the integral signs of Koshє those scho, scho in the members are low є pevna function and її pokhіdna. 3 topics Pokhidna you sing-songly memorized the simplest tabular phrase: , and we ourselves have such a canonical temper.

How to beat the integral sign? On the back we take the icon of the integral and rewrite it with the “little one” in the row of the upper and lower boundaries: . Then, under the integral, we rewrite the “stuffing” next to the letter “he”: . Why don't you pick up ..., oh, so, in the number book it is necessary to stick the pictogram of the differential: .

Now you need to calculate unclassified integral. With this, two fluctuations are possible:

1) If it is clear that the integral converges, then our series also converges.

2) If it is clear that the integral diverges, our series will also diverge.

I repeat, as the material of the launches, the reading of the paragraph will be important and ill-understood, the shards of stosuvanny signs, in fact, can be counted non-consecutive integral first kind.

Outwardly, the solution that the design of the butt can look like this:

Vikoristovuєmo іntegrаlnuyu mark:

In such a rank, doslidzhuvany row disperse at once from the indistinct integral.

butt 12 Continue the row to the left

The decision and the szrazok designed like a lesson

In the case of looking at the butts, the logarithm is also a moment buti under the root, it would change the method of solution.

I have two more for a snack

butt 13 Continue the row to the left

Behind the upper “parameters”, the upper member is low enough to be suitable for the selection of boundary signs of alignment. It’s more necessary to open the shackles and give the candidate a boundary line Danish row zі similar instruction. Vtіm, I'm slyly evil, the bows can and not be twisted, but all the same solution through the boundary sign of alignment looks like a chimera.

Therefore, we have won the integral sign of Cauchy:

The integrand function is uninterrupted on

converge at once from the indistinct integral.

! Note:otrimane number -not є in a row!

butt 14 Continue the row to the left

The solution is that the image is decorated in the end of the section, which is getting the end.

From the method of residual and irreversible assimilation by those numerical rows, see those.

Solution and advice:

Example 3:Victory to the sign of d'Alembert:

In such a rank, doslidzhuvany row disperse.
Note: It is possible to win the “turbo” solution method: circle the setting with an olive, indicate that it is not alone and make a sign: “one order of growth”.

Example 5: Vikoristovuemo sign of d'Alembert: Including, dosledzhuvany row converge.

Example 8:

In such a rank, doslidzhuvany row converge.

Example 10:
Victory to the radical sign of Kosh.

In such a rank, doslidzhuvany row disperse.
Note: Here is the basis of the step, to that

butt 12: Vikoristovuєmo іntegralnomu sign

The last number has been taken away, then the next row converge

butt 14: Vikoristovuєmo іntegralnomu sign
The integrand function is interruptless on .

In such a rank, doslidzhuvany row disperse at once from the indistinct integral.
Note: A number can also be followed for helpborder signs . For this, it is necessary to open the arches under the root and align the rows next to each other to separate.

Rows of signs. Leibniz sign. Apply solution

In order to understand the lesson, it is necessary to be well oriented in positive numerical series: to understand what a series is, to know the necessary sign of profitability is low, to remember to stop the signs of porivnyannia, the sign of d'Alembert, the signs of Cauchy. The topic can be destroyed practically from scratch, sequentially twisting the stats Rows for teapotsі Sign of d'Alembert. Signs of Kosh. It is logical that this lesson is the third one after the rakhunok, and it is allowed not only to rise in the ranks, to know the drawing, but to close the material already passed! If there is no novelty, there will be little, and master the rows that the sign is drawn, not the warehouse of great practice. Everything is just available.

What is a row, what is a row? Tse zroumilo chi mayzhe zrozumilo already from the name itself. Seems like the simplest example. Let's look at a row and write down this report:

And at the same time, there will be a driving comment. In the members of the row that are drawn, the signs are drawn: plus, minus, plus, minus, plus, minus, etc. to inexcusability.
The sign of the devil is a safe multiplier: if it’s a guy, then it’s a plus sign, if it’s not a pair, it’s a minus sign. In mathematical jargon, this contraption is called a "flashing light". In this order, the row that is being drawn is “knowing” by minus loneliness in the step “en”.

In practical butts, the sign of the number of members is low, it can be safe as a multiplier, and yogo should be taken: , , , .... For example:

Under a stone - "tricks":,, etc. - so multipliers do not worry about changing the sign. It dawned on me that I should be natural: , , . A number of tricks are not only especially gifted to students, they sometimes blame "themselves" in the course of the decision functional rows.

How to finish the row, what is being scolded, for profit? Vykoristovuvaty sign of Leibniz. I don’t want to tell anything about the German giant of Gottfried Wilhelm Leibnitz’s thoughts, shards, cream of mathematical practices, wines of a bunch of volumes of philosophy. Not safe for the brain.

Sign of Leibnitz: As a member of a series of alternating characters monotonously fall behind the modulus, then the series converges. Abo has two points:

2) Members are low in modulus: . And why change monotonously.

Yaxcho viconano resentment mind, then the series converge.

A short note about the module is given in the manualHot formulas for school mathematics course , ale for clarity again:

What does "behind the module" mean? Module, as I remember from school, "z'їdaє" sign "minus". Let's go back to the line. Thoughts zіtremo gumkoy all signs i look at the numbers. We care what skin offensive member of a row less lower front. In this order, the next phrases mean the same:

- Member of a row without a badge subside.
- Members of the row change behind the module.
- Members of the row change in absolute terms.
– Module the main member of the series of pragne zero: Kіnets dovidki

Now let's talk a little about monotonicity. Monotony is the price of tedious postiynist.

Members of a row Suvoro monotonously fall behind the module, like a SKIN ADVANCED member in a row behind the module LESS, lower front: . For a number of vikonan, the monotony of the change is strict, and you can report it:

And you can say it in short: leather offensive member in a row behind the module less, lower front: .

Members of a row not strictly monotone subside in modulo, like the SKIN ADVANCED member of the series modulo NOT MORE THAN the front one: . Let's take a look at the series of factorials: Here there is no strict monotonicity, because the first two members of the series are the same for the module. Tobto leather offensive member in a row behind the module no more than the front: .

In the case of the Leibniz theorem, the monotonicity of the fall is to blame (it does not matter, whether it is strict or not). Chomu's members can be low find the actual hour of growth for the module, ale "hvist" in a series of obov'yazkovo can be monotonously falling. You don’t need to slander what I’ve piled up, practical put everything in your place:

butt 1 Continue the row to the left

The last member of the row has a multiplier, which means that it is necessary to win the Leibniz sign

1) The revision is low on the sign of the drawing. Sound at this paragraph of the decision below, write down the report and blame the verdict "Row of signs".

2) What do you change the terms of the series modulo? It is necessary to cross the border, as it is most often even more simple.

- Members of the row do not fall off modulo. Before the speech, there was a need for mirkuvannyah about the monotony of falling. Visnovok: a number of disperse.

How to grow up, why one thing? Kind of simple. As you can see, the module has a minus, so in order to put it together, you just need to clean up a flashing beacon from the daha. At the same time, a hot member of the series. Stupidly pick up the "flashing light": .

butt 2 Continue the row to the left

Vikoristovuemo sign of Leibniz:

1) The series є is in turn.

2) - members of the row fall behind the module. Leather offensive member of the row modulo less, lower front: in this order, the decline is monotonous.

Visnovok: the series converge.

Everything would have been even simpler - but still there is no solution!

If the series converge behind the Leibniz sign, then it also seems that the series converge mentally.

How to converge and series, folding from modules: then it seems that the series converge absolutely.

To that, in the order of the day, another stage of perfection standard task- Vivchennya sign of the next row for absolute comfort.

I'm not guilty - such is the theory of number series =)

Doslіdzhuєmo our row for absolute zbіzhnіst.
We put together a series of modules - again we just pick up the multiplier, which ensures the sign of the drawing: - disperse (harmonious series).

In this manner, our row not absolutely similar.
Next row get smarter.

Respect that in Applied No. 1 it is not necessary to carry out the follow-up of non-absolute profit, the wisps are broken for the first crochet at the one that diverges in a row.

We choose tippers, shovels, cars and, apparently, from the kennel, to marvel at the world with wide flattened eyes from the cab of my excavator:

butt 3 Follow the row to the left Vikoristov's sign of Leibniz:

1)
This series is sign-ordered.

2) - members of the row fall behind the module. Skin advancing member of the series modulo less, lower forward:, also, the decline is monotonous. Visnovok: The series converge.

Analyzing the filling of the row, we come to the visnovka, which is necessary here to vicorate the boundary sign of equalization. The arches at the banner of the banner are better opened:

Porіvnyaєmo given row by row, where to go. Vikoristovuemo boundary sign of the difference.

The end number has been taken away, it has been taken from zero, and the series converge at the same time from next to each other. Next row converge absolutely.

butt 4 Continue the row to the left

butt 5 Continue the row to the left

Tse apply an independent solution. On the outside, the solution is that the design is similar to the distribution.

Like bachite, row, scho swear - it's just boring! Ale, do not hurry to turn the corner, in just a couple of screens we can see the vipadok, which is rich to put at a deaf kut. In the meantime, a couple of examples for training and repetition.

butt 6 Continue the row to the left

Vikoristovuemo sign of Leibniz.
1) The series є is in turn.
2)
Members of the series change after the module. Skin advancing member of the series modulo less, lower forward, same, decline monotonously. Visnovok: the series converge.

Show respect that I have not reportedly painted the members of the row. Їx zavzhdi bazhano rozpisuvati, but in the case of non-separate linoschiv at “important” vipadkas, you can settle with the phrase “Row є sign queue”. Before the speech, it is not necessary to put it formally before this point, for sure(I would like to think) that a number of true signs are drawn. A better look is allowed, and pardon is allowed “on automatic”. Remember about "tricks", if they stink, then you need to get rid of them, otrimavshi "superior" series of positive members.

Another subtlety is worth phrases about monotony, її tezh as quickly as possible. So it’s possible to work, and may you wait for your task to be buried. I’ll tell you the hell out of it - especially I often start talking about monotony and such a number to pass. Ale, be ready to paint in detail, right up to the report lances of irregularities (divine butt on the cob of the lesson). In addition, sometimes the monotony becomes suvoroy, and for this, there are steps to replace the word “less” with the word “no more”.

Doslіdzhuєmo a number of absolute zbіzhnіst:

It is obvious that it is necessary to victorize the radical sign of Kosh:

In such a rank, it is low to converge. Next row converge absolutely.

butt 7 Continue the row to the left

This is an example for an independent vision. Quite often the rows are chirping, which the chergs signify, like the call of difficulties.

butt 8 Continue the row to the left

Vikoristovuemo sign of Leibnitz:
1) The series є is in turn.

On the right, in that there are no standard primordial methods for executing such inter. Kudi pragne such a boundary? To zero, to insincerity? It's important here- numeral chi banner.

NOTE: the understanding of the order of growth of the function is reportedly displayed in the articleMethods of rozvyazannya between . We have between sequences but do not change the essence.

As a numeral with growth rate for factorial, then . Yakshcho, on the inconsistency of the factorial grows faster than the numeral, vin, navpaki - “tightening” the boundary to zero: . And can it be intermediary to a certain number like zero?

Let's try to write down the number of first members is low:
you can imagine the rich term of the thousandth step, but I won’t change the situation again - early on, the factorial has all the same “experienced” and such a terrible rich term. Factorial more high order growth station nizh be-like a stately sequence.

– Factorial height swidshe, lower dobutok be-like a kіlkostі ostentatious and static sequences (our type).

– Be a Yaka showing the sequence is growing faster, lower, be a static sequence, for example:,. Display sequence in the highest order of growth nizh be-like a stately sequence. Similar to the factorial, it shows the sequence of “pulling” the addition of any number of whether there are any number of sequences or rich sections: .

- And what is "cooler" factorial? Є! Step-showing sequence ("en" at the step "en") grows faster than the factorial. It is true that it is rarely heard, but the information of the application is nothing to be cleared up. Kіnets dovidki

In this order, another item of follow-up (do you remember more? =)) can be written as follows:
2), shards of the highest order of growth, lower.
The members of the series change after the module, starting from the current number, with this, the skin offensive member of the series modulo less, lower forward, in such a rank, the decline is monotonous.

Visnovok: the series converge.

The axis here is just that cicavia vipadok, if the members of the row of the head grow modulo, through which we have formed a pardon for the first thought about the boundary. ale, starting from the deyakogo number "en", the factorial is going down the numeral, and the “tail” in the series becomes monotonously decreasing, which is important to the mind of Leibniz’s theorem. Why itself and tse "en", it's difficult to find out.

According to the obvious theorem, from the absolute efficiency of the series, the apparent and mental efficiency of the series. Visnovok: Next row converge absolutely.

І nasamkinets a couple of applications for an independent solution. One of the tієї zh opera (reread the conclusion), ale simpler. The second for gourmets - on the fixed integral signs of zbezhnosti.

butt 9 Continue the row to the left

butt 10 Continue the row to the left

After a similar interpretation of numerical positive and familiar rows, from pure conscience, you can go to functional rows, yakі not less monotonous and odnomanіtnі cіkavі.

Solution and advice:

butt 4: Vikoristovuemo to Leibniz's sign:

1) This series is sign-alternative.
2)
Members of the row do not fall behind the module. Visnovok: A number of disperse.. , with this, the skin offensive member of the series modulo less, lower forward, in such a rank, the decline is monotonous.

In this order, the series diverge at once from the inconsistent integral. Next row get smarter.

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 .

I'll put it in number series you can add the sum of the infinitely decreasing geometric progression with the 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: .

One more butt of a rozbіzhny numerical series . At tsomu vipadku n-a chastkova sum can be counted as yak. The border between private sums is unskinned .

Suma mind called harmonic numbers nearby.

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 harmonic 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.

When victorious sufficient sign In order to continue the numerical rows on the scale, it is always necessary to roll over, so it is recommended to roll up to the first 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 selected as the indicator of the step of the th k-th member of the cost of indicators in the step 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. Well, if the unsettled integral diverges, the outer row can also diverge.

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.

Before that, how to formulate the sign itself, a look at an important food:
If it is necessary to zastosovuvat sign of d'Alembert's wealth?

The main rethinking signs of d'Alembert are as follows:

1) Before the final member of the series (“stuffing” of the series), enter the same number in the steps, for example,, and so on. Moreover, it’s not important, it’s not important, where the functions are roztashovuetsya, it’s important for the numeral officer or the bannerman, that the stench is present there.

2) Factorial enters before the start member of the series. What is factorial?

…

…

! When vikoristanny signs of d'Alembert, we happen to write the factorial report. Like i in the front paragraph, the factorial can be spread in the mountains or below the fraction.

3) Like in the zagalny member of the series є "languide of multipliers", for example, . Tsej vpadok trapleyaetsya rarely.

Together with the steps of chi (i) factorials in the beginning of the series, rich terms often grow, but it doesn’t change right - it’s necessary to win the sign of d’Alembert.

Without rozuminnya between that vminnya rozkrivat invisibility far away, unfortunately, do not stick out.

Butt:
Solution: Mi, scho in the sleeping member is low, but it is correct to reconsider those who need to win the sign of d'Alembert.

Victory to the sign of d'Alembert:

converge.

Radical sign of Kosha.

The sign of zbіzhnosti Koshі for positive numerical rows, in addition to the surprisingly well-looked-for sign of d'Alembert.

! It means that if the sign of Cauchy does not give us advice on food about the economy in a row, then the sign of d'Alembert will not give us advice. But if the sign of d'Alembert is given, then the sign of Kosh can be "repaired" as a whole. Tobto is a sign of Koshі є in tsomu sensі a strong sign.

!!! When is it necessary to vicorate the radical sign of Kosh? Radical signs of Kosh sound vikoristovu in quiet vipadkah, if a hot member is in a row POVNISTTYU to be in the step lie down like "en". Abo if the root of "good" rises from the hard member is low. There are more exotic vipads, but they don’t clog your head with them.

Butt: Continue the row to the left

Solution: Mi Bachimo, that the main member of the series is constantly changing under the step, which is to fall in, then, it is necessary to win the radical sign of Kosh:

In such a rank, doslidzhuvany row disperse.

Integral sign of Cauchy.

In order to establish the integral sign of Kosh, it is necessary to have a greater or lesser amount of knowledge about the difference, the integration, and also the mother of the calculation non-consecutive integral first kind.

I will formulate in my own words (for the sake of simplicity, I will explain).

Integral sign of Koshі: Look at positive number series. This series converge and diverge at the same time from a variable unsettled integral.

! !! The main rethinking of the integral signs of Koshє those scho, scho in the members are low є pevna function and її pokhіdna.

Butt: Continue the row to the left

Solution: 3 topics Pokhidna you sing-songly memorized the simplest tabular phrase: , and we ourselves have such a canonical temper.

Now it is necessary to virahuvat the unclassified integral. With this, two fluctuations are possible:

1) If it is clear that the integral converges, then our series also converges.

2) If it is clear that the integral diverges, our series will also diverge.

Vikoristovuєmo іntegrаlnuyu mark:

The integrand function is uninterrupted on

In such a rank, doslidzhuvany row disperse at once from the indistinct integral.

Butt: Continue zbіzhnist row

Solution: we are going to be reconsidered a necessary sign of comfort in a row. This is not a formality, but a miraculous chance to get rid of the butt with “little blood”.

Numerical sequence viscogo in order chim, that , so it is necessary for a sign of zbіzhnostі vikonano, and a row can, like converge, and diverge.

In such a rank, it is necessary to victorious, be it a sign. What a hello? Boundary sign clearly do not fit, the logarithm wormed its way into the upper member of the row, signs of d'Alembert and Kochi so do not lead to a result. We have a yakby buv, then so-and-so it’s possible to turn around through integral sign.

“Looking at the place below” to suggest a row, which should diverge (a drop of a knotted harmonic row), but again I blame the food, how to cheat the logarithm of the number book?

The first sign of a break, foundations on inconsistencies, is lost, as often not taken to respect and falls with a saw on a distant police. Let's write a number of reports:

Guess what - uncircumcised growing numerical sequence:

І, starting from the number, the inconsistency will be fixed:

tobto, the members of the series will be even more members retail row.

The results of the series do not have anything left, just as they diverge.

Skhіdnіst chi rozbіzhnіst numeric series lie in the yogo "unskinned tail" (surplus). In our opinion, we can not take to respect the fact that the unevenness is wrong for the first two numbers - it doesn’t add up to the splitting of the visnovoks.

A neatly designed butt can look something like this:

Porіvnyaєmo tsey row z next, scho disperse.
For all numbers, starting from, vikonano nerіvnіst , then, for the sign of the order of the next disperse.

Rows of signs. Leibniz sign. Apply roses.

What is a row, what is a row? Tse zroumilo chi mayzhe zrozumilo already from the name itself. Seems like the simplest example.

Let's look at a row and write down this report:

The black sign is a safe multiplier: if it’s a guy, then it’s a plus sign, if it’s not a pair, it’s a minus sign

In practical butts, the sign of the number of members is low, it can be safe as a multiplier, and yogo should be taken: , , , .... For example:

Under a stone - "tricks":,, etc. - so multipliers do not worry about changing the sign. It dawned on me that I should be natural: , , .

How to finish the row, what is being scolded, for profit? Vykoristovuvaty sign of Leibniz.

Sign of Leibnitz: Like in a row, what is drawn, two minds are counted: 1) the members of the row monotonously fall behind the absolute value. 2) the interim term after the modulus reaches zero, then the series converges, and the modulus of the summation series does not shift the modulus of the first term.

A short summary about the module:

What does "behind the module" mean? Module, as I remember from school, "z'їdaє" sign "minus". Let's turn to the row . Thoughts zіtremo gumkoy all signs i look at the numbers. We care what skin offensive member of a row less lower front.

Now trochs about monotony.

Members of a row Suvoro monotonously fall behind the module, like a SKIN ADVANCED member in a row behind the module LESS, lower front: . For a row the strict monotony of the decline was vikonan, її it is possible to write a report:

And you can say it in short: leather offensive member in a row behind the module less, lower front: .

Members of a row not strictly monotone subside in modulo, like the SKIN ADVANCED member of the series modulo NOT MORE THAN the front one: . Let's look at a series of factorials: Here there may be a non-strict monotonicity, because the first two members of the series are the same for the module. Tobto leather offensive member in a row behind the module no more than the front: .

Butt: Continue the row to the left

Solution: The last member of the row has a multiplier, which means that it is necessary to win the Leibniz sign

1) Rechecking the series on a monotonous change.

1<2<3<…, т.е. n+1>n persha umova do not win

2) - Umov's friend is not a vikonan.

Visnovok: a number of disperse.

Appointment: If a series converge after the Leibniz sign and a series, foldings from modules: also converge, then it seems that a series converge absolutely.

If a series converge after the Leibniz sign, and a series of foldings from modules: to diverge, then it seems that a series converge mentally.

If a series of foldings from modules converge, then the whole series converges.

To that, a sign of blackness, a similar series is necessary to reach absolute or mental feat.

Butt:

Solution: Vikoristovuemo sign of Leibniz:

1) The skin advancing member in a row behind the module is smaller, lower in front: - the first Umov vikonan.

2) - Umov's friend is also a vikonan.

Visnovok: the series converge.

Let's change it to mindfulness and absolute comfort.

We put together a number of modules - we just pick up the multiplier again, which ensures the sign of the drawing:
- Disperse (harmonious series).

In this manner, our row not absolutely similar.
Next row converge mentally.

Butt: Follow the row to the mindfulness of the absolute economy

Solution: Vikoristovuemo sign of Leibnitz:
1) Let's try to write down the number of first members is low:

…?!

On the right, in that there are no standard primordial methods for executing such inter. Kudi pragne such a boundary? To zero, to insincerity? It's important here- numeral chi banner.

As a numeral with growth rate for factorial, then . For example, on the inconsistency of the factorial growth, the growth is faster than the number one, then vin, navpaki - “pulling” the boundary to zero: . And can it be intermediary to a certain number like zero? or . You can substitute the same rich term of the thousandth step, but I don’t want to change the situation again - sooner or later factorial all the same “experienced” and such a terrible rich term. Factorial in the highest order of growth.

– Factorial height swidshe, lower dobutok be-like a kіlkostі showy and static sequences(Our vipadok).

– Be a Yaka showing the sequence is growing faster, lower, be a static sequence, for example:,. Display sequence in the highest order of growth no matter how static sequence. Similar to the factorial, it shows the sequence of “pulling” the additional number of whether there are any number of stacked sequences or rich segments: .

– And what is “stronger” for factorial? Є! Step-showing sequence ("en" at the step "en") grows faster for factorial. It is true that it is rarely heard, but the information of the application is nothing to be cleared up.

Kіnets dovidki

In this order, another point of follow-up can be written as follows:
2) , shards of a higher order of growth, lower .
The members of the series change after the module, starting from the current number, with this, the skin offensive member of the series modulo less, lower forward, in such a rank, the decline is monotonous.

Visnovok: series converge

The axis here is just that cicavia vipadok, if the members of the row of the head grow modulo, through which we have formed a pardon for the first thought about the boundary. ale, starting from the deyakogo number "en", the factorial goes around the numeral, and the “tail” becomes monotonically decreasing in a row, which is fundamentally important for understanding Leibniz’s theorem. Why itself and tse "en", it's difficult to find out.

Doslіdzhuєmo a number of absolute chi umovnu zbіzhnіst:

And here the d'Alembert sign is already practiced:

Victory to the sign of d'Alembert:

In such a rank, it is low to converge.

Next row converge absolutely.

Razіbraniya butt can be vyrishiti in another way (vykoristuєmo suffices for a sign of zbіzhnosti row, scho to be drawn).

Sufficient sign zbіzhnostі sign of an alternating row: If a series of foldings from the absolute values of the members of this series converge, then the whole series converges.

Other way:

Follow the row to the mindfulness of the absolute economy

Solution : Doslіdzhuєmo a number of absolute zbіzhnіst:

Victory to the sign of d'Alembert:

In such a rank, it is low to converge.
3 sufficient signs of zbіzhnostі sign alternating series, converge and series.

Visnovok: Next row converge absolutely.

To calculate the sum to a series of given accuracy we will beat the advent of the theorem:

Let the row, what are you swearing, row pleasing the minds of the signs of Leibniz and let it go - yoga n private sum. Then the series converge and swear at the nearest calculation of the sum S in absolute value, do not change the modulus of the first dropped term:

Functional lava. Steps in a row.
Area zbіzhnostі row.

For the successful mastering of those, it is necessary to be well educated on the most significant numerical series.

Signs of living rows.
Sign of d'Alembert. Signs of Kosh

Practice, practice - and we'll come later
J.L. d'Alembert

Usikh vіtayu with the cob of the initial rock! Today is the 1st of Spring, and I believe in the honor of the holy cognizance of the readers of this, that you have long been counting with impatience and stumbled upon the recognition signs of success of numerical positive rows. Holy First Spring, that my welcome is still relevant, nothing terrible, as if it’s true for the weekend, see if you meet again at the same time, you’ll sleep, learn, as if you went to the right side!

For those who are only just beginning to turn a row, I recommend that you read this article from the beginning Numerical rows for teapots. Vlasne, tsey vіzok є prodovzhennyam banquet. Otzhe, today at the lesson, we can look at this solution to that:

One of the widest signs of porivnyannya, which is in practical butts, is the sign of d'Alembert. The signs of Kosh are more familiar, but also more popular. As always, I will try to compose the material simply, it is understandable and understandable. The theme is not the most complicated, and all the tasks of the singing world are stenciled.

The Sign of the Life of d'Alembert

Before that, how to formulate the sign itself, a look at an important food:
If it is necessary to zastosovuvat sign of d'Alembert's wealth?

1) The bannerman has a rich member.
2) Rich terms are found in the number book, and in the banner.
3) One or the other richly divided can be rooted.
4) Richly articulated and rooted, wisely, maybe more.

The main rethinking signs of d'Alembert are as follows:

1) Before the final member of the series (“stuffing” of the series) enter as a number in the steps, for example,,, and so on. Moreover, it’s not important, it’s important, where the contraption is roztashovuetsya, it’s important to the numeral officer or the bannerman, it’s important that it’s there.

2) Factorial enters before the start member of the series. With factorials, we crossed swords at the lesson. Numerical sequence and її boundary. Vtіm, do not make a new tablecloth-self-assembly:

…

…

! When vikoristanny signs of d'Alembert, we happen to write the factorial report. Like i in the front paragraph, the factorial can be spread in the mountains or below the fraction.

3) Like in the zagalny member of the series є "languide of multipliers", for example, . This vapadok trapleyaetsya rarely, ale! When such a series is completed, pardons are often allowed - divas. Example 6.

Together with the steps of chi (i) factorials in the beginning of the series, rich terms often grow, but it doesn’t change right - it’s necessary to win the sign of d’Alembert.

Who has problems between the borders, or unreasonable between, back to the lesson Mezhі. Apply solution. Without rozuminnya between that vminnya rozkrivat invisibility far away, unfortunately, do not stick out.

And now apply the dovgoochіkuvani.

butt 1

Mi, scho in the sleeping member is low, but it is correct to reconsider those who need to win the sign of d'Alembert. On the other hand, the decision is made, the commentary is below.

Victory to the sign of d'Alembert:

converge.
(1) We add the position of the offensive member to the anterior one: . Mind bachimo, scho is a hot member of the series. In order to take away the offensive member, it is necessary for a number of DEPUTY submit: .
(2) Let's take some extra surface shot. For the singing of the decision, you can skip this part.
(3) The arms of the dialer are curved. At the bannerman, we blame the fourth step.
(4) Fast on . We blame the constant for the boundary. At the numeral in the bows, similar dodanki are induced.
(5) Insignificance is established in the standard way - by subdividing the numeral and banner into "en" at the senior level.
(6) Divide the numerals into banners one by one, and it is possible to make additions, so as to overcome zero.
(7) Let's just follow and work on the sign, that with a whisker about those who follow the sign of d'Alembert, a series of converge.

In the open butt, in the deep member, we have a low, rich member of the 2nd degree. What work, how is there a rich member of the 3rd, 4th or higher level? On the right, in the fact that the rich term of the higher level is given, then it is difficult to blame the opening of the bow. In this way, you can zastosovuvat "turbo"-method of solution.

butt 2

Let's take a similar series and do it for profit

Victory to the sign of d'Alembert:

In such a rank, doslidzhuvany row converge.

(1) We make a deposit.

(3) Look at Viraz at the clerk, that viraz at the bannerman. My bachimo, that in the number book it is necessary to open the arches and lead to the fourth step: what do you want to work with? And for those who do not know Newton's binomial, it will appear even more foldable. Let's analyze the senior steps: as if to the beast, we'll open the arches , then we take away the senior step. Below we have such the most senior step:. By analogy with the front butt, it is obvious that we have a mezhіde loneliness with the punctuated rozpodіlі of the numeral and banner. Abo, as mathematicians seem, richly articulated і – one order of growth. In such a rank, you can circle the shutter let's forgive the oliver and say once again that this thing is not alone. Similarly, we deal with another pair of rich members: and stink also one order of growth, that yoga setting is pragne alone.

Indeed, such a “hack” could be done in Example No. 1, but for a rich member of the 2nd stage, such a decision still looks undignified. Especially I work like this: if it is a rich member (or a rich member) of the first or another stage, I use the vicorist “long” method of opening Butt 1. This is how the rich member of the 3rd and more high steps is traversed, I use the vicorist “turbo” method on the upper butt 2.

butt 3

Continue the row to the left

Let's take a look at typical applications with factorials:

butt 4

Continue the row to the left

Before the zagalny member of the series, enter і steps, і factorial. It dawned on me, like daylight, that the sign of d'Alembert needed to be victorious here. We see it.

In such a rank, doslidzhuvany row disperse.
(1) We make a deposit. Let's repeat it one more time. Behind the mind is a burning member of the row: . To take away the offensive member of the row, deputies need to provide, in this manner: .
(2) Let's take some extra surface shot.
(3) Vіdshpuєmo simku vіd step. Factorials are written in a report. Yak tse zrobiti - div. the beginning of the lesson or the article about the number sequence.
(4) Everything that can be shortened is fleeting.
(5) The constant is blamed for the boundary sign. At the number book, the arches are crooked.
(6) Non-significance is assessed in the standard way - by subdividing the numeral and banner into “en” at the senior level.

butt 5

Continue the row to the left

Outwardly, the solution is that the lesson is designed like a lesson

butt 6

Continue the row to the left

From the layout of mi bachimo, that at the skin offensive member of the row, an additional multiplier is added to the bannerman, that is, as a hot member of the row , then the next term of the series:
. The axis here is often automatically pardoned, formally recording after the algorithm what

A clear eye of decision may look like this:

Victory to the sign of d'Alembert:

In such a rank, doslidzhuvany row converge.

Radical sign of Kosh

The sign of zbіzhnosti Koshі for positive numerical rows, in addition to the surprisingly well-looked-for sign of d'Alembert.

Radical sign of Koshі: Look at positive number series. Yakscho є boundary: then:
a) At a row converge. Zokrema, the series converge for .
b) At a row disperse. Zokrema, a row to disperse for.
at Pri the sign is not given. It is necessary to win another sign. It means that even if the sign of Kosh does not give us a confirmation on the food about the economy is low, then the sign of d'Alembert will not give us a confirmation. But if the sign of d'Alembert is given, then the sign of Kosh can be "repaired" as a whole. Tobto is a sign of Koshі є in tsomu sensі a strong sign.

When is it necessary to vicorate the radical sign of Kosh? The radical sign of Kosh sings victorious in vipadkas, if the root “good” emerges from the main member of the row. As a rule, the whole pepper is rebuffed in stages, yak lie down. There are more exotic vipads, but they don’t clog your head with them.

butt 7

Continue the row to the left

Mi Bachimo, scho to be more familiar with the step, which is to fall in the form of "en", and also, it is necessary to win the radical sign of Kosh:

In such a rank, doslidzhuvany row disperse.

(1) We make out the main member of the series of sub roots.

(2) We rewrite the same ones, only already without a root, victorious power of steps.
(3) At the showman, we divide the number by term for the banner, indicating that
(4) As a result, we have become insignificant. Here you can drink for a long time: call into a cube, call into a cube, then divide the number and banner into "en" at the cube. But in this way, there is an effective solution: by this method you can win right under the step-constant. For usunennya non-significance, we divide the numeral and banner into (the senior step of the richly divided).

5
(6) It is brought to a path, we note that we are robimo visnovok about those who are in a row to disperse.

And the axis is a simple butt for an independent vision:

butt 8

Continue the row to the left

І more sprat of typical applications.

Outwardly, the solution is that the lesson is designed like a lesson

butt 9

Continue the row to the left
Vikoristovuemo radical sign Kosha:

In such a rank, doslidzhuvany row converge.

(1) We place the head member in the row of root.

(2) We rewrite the same, but already without a root, with which we open the arches, the vicorist formula of the shortened multiplication: .
(3) At the pokanik, we divide the number by term for the znamennik, and it is indicated that.
(4) The insignificance of the mind has been taken away, and here it is also possible to win the rose right under the step. Ale with one mind: the coefficients at the senior levels of richly articulated can be different. We have a stench of different (5 and 6), and for this it is possible (and necessary) to divide insults on top of. What are the coefficients however, for example (1 and 1): then such a trick should not be passed and it is necessary to twist another miraculous boundary. As you remember, the subtleties were seen in the rest of the article Methods of rozvyazannya between.

5
(6) Insignificance has been usuned, we have lost the simplest boundary: . Why in infinitely great step pragna to zero? Because the basis of the step is satisfied with the nervousness. How can someone have doubts about justice between , then I won’t hesitate, I’ll take a calculator in my hands:
Yakscho something
Yakscho something
Yakscho something
Yakscho something
Yakscho something
… and so on. to inexcusability - tobto, between:

Just the same infinitely decreasing geometric progression on fingers =)
! Don't beat this trick like a proof! Because it's more obvious, it still doesn't mean it's right.

(7) It is possible, how and robimo visnovok, to the one that the series converges.

butt 10

Continue the row to the left

This is an example of an independent solution.

Integral sign of Koshi

Abo is just an integral sign. I rejoice quietly, who has rottenly acquired the material of the first course. In order to establish the integral sign of Kosh, it is necessary to have a greater or lesser amount of knowledge about the difference, the integration, and also the mother of the calculation non-consecutive integral first kind.

At assistants for mathematical analysis integral sign of Cauchy it is given mathematically suvoro, but it’s already confused, then I will formulate a sign that it’s not suvoro, but it’s understood:

Look at positive number series. If there is an inconsistent integral, then the series converge and diverge at the same time from the integral.

I just apply for clarification:

butt 11

Continue the row to the left

Mayzhe classic. Natural logarithm and as a bad thing.

The main rethinking of the integral signs of Koshє those that have low multipliers in the common member, similar to the deak function and її pokhіdnu. 3 topics

d'Alembert's sign of safety Radical sign of safety of Kosh

Before that, how to formulate the sign itself, a look at an important food:
If it is necessary to zastosovuvat sign of d'Alembert's wealth?

We'll start from repetition. Let's guess the ups and downs, if it's necessary to stop the finds boundary sign. The boundary sign of the alignment is stagnant only if the main member of the row has:
1) The bannerman has a rich member.
2) Rich terms are found in the number book, and in the banner.
3) One or the other richly divided can be rooted.

The main rethinking signs of d'Alembert are as follows:

2) Factorial enters before the start member of the series. From factorials we crossed swords at the lesson. Numerical sequence and її boundary. Vtіm, do not make a new tablecloth-self-assembly:

…

…

! When vikoristanny signs of d'Alembert, we happen to write the factorial report. Like i in the front paragraph, the factorial can be spread in the mountains or below the fraction.

Together with the steps of chi (i) factorials in the beginning of the series, rich terms often grow, but it doesn’t change right - it’s necessary to win the sign of d’Alembert.

In addition, the primal member of the series may have one hour at a time and a step and a factorial; you can use two factorials, two steps, it’s important that there was want iz looked at points - and tse rethought signs of d'Alembert.

Sign of d'Alembert: Look at positive number series. How is it between the extension of the offensive member to the anterior one: then:
a) At a row converge. Zokrema, the series converge for .
b) At a row disperse. Zokrema, a row to disperse for.
at Pri the sign is not given. It is necessary to win another sign. It’s more common to go alone at that vapadka, if the signs of d’Alembert are trying to get stuck there, it’s necessary to beat the boundary sign of equalization.

And now apply the dovgoochіkuvani.

butt 1

Mi, scho in the sleeping member is low, but it is correct to reconsider those who need to win the sign of d'Alembert. On the other hand, the decision is made, the commentary is below.

Victory to the sign of d'Alembert:

converge.

butt 2

Let's take a similar series and do it for profit

Victory to the sign of d'Alembert:

In such a rank, doslidzhuvany row converge.

butt 3

Continue the row to the left

Outwardly resolved and completed in the end of the lesson about the numerical sequence.
(4) Everything that can be shortened is fleeting.
(5) The constant is blamed for the boundary sign. At the number book, the arches are crooked.
(6) Non-significance is assessed in the standard way - by subdividing the numeral and banner into “en” at the senior level.

butt 5

Continue the row to the left

Outwardly, the solution is that the lesson is designed like a lesson

butt 6

Continue the row to the left

A clear eye of decision may look like this:

Victory to the sign of d'Alembert:

In such a rank, doslidzhuvany row converge.

Follow the vicarious row, vicorist to the sign of d'Alembert. Numerical rows: appointment, power, signs of wealth, butt, decision. Sufficient signs of life in a positive sign series

The main purpose is to understand.

The power of similar numerical series.

The necessary mental health is low.

Sufficient signs of zbіzhnostі sign-positive series.

That sufficient mindfulness of a positive-sign numerical series is necessary.

The first, the other, and the third signs are equal.

Sign of d'Alembert.

Radical sign of Kosha.

Integral sign of Cauchy.

Signs of living rows. Sign of d'Alembert. Signs of Kosh

The Sign of the Life of d'Alembert

Radical sign of Kosh

Integral sign of Koshi

Signs of living rows.
Sign of d'Alembert. Signs of Kosh