Perform a forward check. It’s a simple theorem to prove that if the sum of a function f converges, then the sum of the functions f is equal to 0. Also, since we have a function x^2, there are boundaries, and the sum of them diverges infinitely ; On the other hand, between functions 1/x equals 0, then their sums can converge. Since the boundary is not equal to zero, we know that the series diverges. RESPECT: it’s not wrong, because those between the two equal zeros do not mean at all that the series converges obligatorily. Whose problem requires further re-verification.
Geometric rows. For these rows there is a very simple rule, which is most important because your row is not geometric. A geometric series is a sequence of numbers, each of which can be seen as r^k, where k is changeable, and r is a number that lies in the interval between -1 and 1. Geometric series always converge. Moreover, you can easily calculate the sum of a series similar to 1/(1-r).
Uralized harmonious series, like Dirichlet series. Such a series is called a sum of a function of the form 1/(x^p), where x is a number. The theorem for these series states that if p is greater than one, the series will converge, and if p is less than one, the series will diverge. This means that most of the guessing series 1/x diverge, its fragments can be as 1/(x^1), where p=1. This series is called harmonic. The series 1/(X^2) converges, fragments 2 are greater than 1.
Other love. If the row does not fit up to one of the types designated more, draw to a new method, move it lower. If one method did not work, stop the steps, so that it is not always clear which track to choose. Although there are no unambiguous rules, you can now better navigate the choice of the required method.
- The equalization method. Let’s say you have two rows that are made up of positive terms, a(n) and b(n). Then: 1) if the sum b(n) converges, and the sum a(n) is smaller than b(n) (for whatever it is, the sum of the great n), then the sum a(n) also converges; 2) if b(n) diverges, and a(n)>b(n), then a(n) also diverges. For example, you have a row of 2/x; We can equalize it with 1/x. As we know, the row 1/x diverges, and 2/x > 1/x, so it means that the row 2/x also diverges. Well, the idea of the method is to determine, converge, or follow up the series, the vikorist series, or the previous series.
- Method of equalization between. Since a(n) and b(n) are series of positive numbers, and since there is a boundary between a(n)/b(n), which is greater than 0, then the two rows either converge or diverge. And here a number of investigations are also similar to what we know; The method is to select the next row, the maximum step of which corresponds to the step of the next row. For example, if you look at the row 1/(x^3+2x+1), it makes sense to align it with the row 1/(x^3).
- Verification by integral. Since the function is greater than zero, is continuous and changes for values of x greater than or equal to 1, then the infinite series f(n) converges, like the integral of 1 to infinity with the function f(x) The final values begin and end; Otherwise the series will diverge. Thus, it is enough to integrate the function and find the boundary at x, which goes to infinity: if the boundary is completed, the series converges, if between the previous inconsistencies, the series diverges.
- Significant rows. If a(k)>a(k+1)>0 when there are great k and between a(n) is equal to 0, then the signed series (-1)^n a(n) converges. To put it simply, it is acceptable that your row is significant (its members are alternately positive and negative); In this case, add the significant part of the function and find between what is missing - as between the ends, the series converges.
- Method of marriage. Given an unfinished series a(n), find the next term in the series a(n+1). Then calculate the ratio of the front member to the front one a(n+1)/a(n), taking into account its absolute value. Find out the relationship between n and infinitely; If this is between the beginning and the end, it means advancing: 1) if there is less than one between, the series converges; 2) if there is more than one between them, the row diverges; 3) if between ancient units, this method is insufficient (a series can converge and also diverge).
- These are the main methods for determining the softness of the rows and the stench is extremely brown. As long as I didn’t care about them, it’s absolutely incredible that the matter has no solution, but here there have been arrears. These methods can also be used for other series, such as statistical series, Taylor series, etc. It is difficult to overestimate the profitability of these methods, but other simple methods of calculating the profitability are low.
It is only possible to calculate the sum of a series once the series converges. As soon as the series diverges, the sum of the series is infinite and there is no sense in calculating. The results of the practice show that the sums are low, as stated at the Lviv National University named after Ivan Franko. The order is to select the rows so that the mental capacity is eliminated, if we check for the ability we will be eliminated. These are the statistics behind it that are based on the results of control work based on the analysis of series.
Butt 1.4 Calculate the sum of the rows:
A)
Calculation: Remnants between the frontal member of the row with the advancing number to infinity before 0 
then this series converges. Calculate sums are low. For this purpose, we can transform the hidden member by breaking it down into the simplest fractions of type I and type II. The method for decomposing simple fractions will not be outlined here (it is better described when integrating fractions), but rather we will write down the end type of decomposition 
It is obvious to what extent the sum can be written through the sum of the series created from the simplest fractions, and then from the difference in the sums of the series 
Next, we paint the skin row in a clear bag and see the additions (armchairs), which are transformed 0 after folding. In this way, the sum of the row will be reduced to the sum of 3 dodanki (marked as black), which will result in 33/40. 
This is what the entire practical part of finding money for simple lavas is based on.
Buttons on the folding rows are reduced to the endless progress of those rows, which can be found through similar formulas, but here such butts are not visible.
b) 
Calculation: Known between the nth member of the sumi 
Vaughn is equal to zero, from now on the number of tasks converge and there is a sense of searching for his sum. If the difference is equal to zero, then the sum is a number of ancient inconsistencies with a plus or minus sign.
We know the sum is low. For which the first member of the series is a fraction that can be resolved by the method of insignificant coefficients to the sum of simple fractions of type I 
Following the instructions given earlier, we write down the sum of the series through the subordinate sums of the simplest fractions. 
We are writing the sums and apparently the additions that will become equal to 0 under the hour of subsumption. 
As a result, we deduct the amount of how many additional money (seen in black), which is equal to 17/6.
Example 1.9 Find the sum of the row:
A)
Enumeration: Enumeration to the cordon 
Let's reconfigure so that this series converges and you can find the sum. Next, the sign of the function from the number n is expanded into simple multipliers, and the entire fraction is converted into the sum of simple fractions of type I 
Next, the amount of the row is assigned to the layout and is written down in two simple steps 
The series is written in a clear manner and it is visible that additions are made, which after addition give the sum zero. Other warehouses (shown in black) and the end row 
In this manner, in order to know the amount of a row, you need to put 3 simple fractions under the grave banner.
b) 
Calculation: Cordon of a member of a row with great values of the number of pragne zero 
Why does it matter that the series converges, and why does the sum end? We know the sum of the series, for which, using the method of insignificant coefficients, we divide the first member of the series into three simplest types 
Obviously, the sum of a row can be converted into the sum of three simple rows 
Then we look at the warehouses of all three sums, which after summation turn to zero. In rows containing three simple fractions, one of them, when added, becomes equal to zero (see reds). This is a kind of hint for calculations 
The amount in the row is equal to the amount of 3 additional donations and equal to one.
Stock 1.15 Calculate the sum to the row:
A) 
Calculation: With a halal member of the row, it goes to zero 
this series converges. Let's reshape the sleeping member in such a manner as to make a sum of the simplest fractions 
The range of tasks in a row, based on the layout formulas, is written through the sum of two rows 
After recording in an explicit manner, the majority of terms in the series as a result are assumed to be equal to zero. Forfeit the sum of three donanks. 
The sum of the number series is equal to -1/30.
b) 
Calculation: The fragments between the frontal member are close to zero, 
then the series converges. To find out sumi, we decompose the literal term into fractions of the simplest type. 
During the hour of unraveling, the method of insignificant coefficients was used. We write down the amount in the row of the found layout 
In the near future, it appears that the donations will not make a significant contribution to the final sum and other 
The amount is comparable to 4.5.
Stock 1.25 Calculate the sum of the rows:
A) 
If the fragments are equal to zero, then the series converges. We can know the amount is low. For which, behind the diagram of the front butts, we lay out the front member of the row through simple fractions 
This allows you to write down a series through the sum of simple series i, having seen in the new additions, having asked for this summation. 
In this case, you will lose one additional item like one of the ancient units.
b)
Calculation: We know the cordon of the sleeping member in a row 
And we reconfigure until the series converges. Next, the final term of the number series is decomposed into fractions of the simplest type using the method of insignificant coefficients. 
Using the same fractions we write the sum in the series 
We write down the rows in an obvious way and can be simplified to a sum of 3 additions 
The amount is equal to 1/4.
At this point, the understanding of the schemes for summing up the rows is completed. Here we have not yet considered the series that lead to an endlessly declining geometric progression, which replaces factorials, static patterns, and the like. However, the material will be useful for students on tests and tests.
Number series. Similarity and divergence of number series. D'Alembert sign of danger. Significant rows. The mentality of the rows is absolute. Functional lava. Step rows. Layout of elementary functions in a row Maclaurin.
Methodical passages on topic 1.4:
Number series:
The number next to it is called the sum of the mind
de numbers u 1, u 2, u 3, n n, they are called members of a series, they create an endless sequence; the un member is called the outer member of the series.
. . . . . . . . .
combinations from the first members of the series (27.1) are called private sums of this series.
The sequence of partial sums can be equalized to the skin row S 1, S 2, S 3. When the number n of the private sum increases in a row S n Pragne to the limit S, then the series is called similar, and the number S- a bag of a similar row, then.
This record is equivalent to the record
Yakshcho chastkova sum S n series (27.1) with unbounded growth n there is no completed boundary (zokrema, pragne to + ¥ or to - ¥), then such a row is called a split
If the series converges, then the meaning S n when dosit great n є close viraz sumi row S.
Retail r n = S - S n is called too row. If the series converges, then the surplus will be zero, then. r n = 0, and by the way, if there is too much value of zero, then the series converges.
A number of species are called geometrically nearby.
called harmonious.
yakscho N®¥, then S n®¥, then. the harmonic series diverges.
Example 1. Write down a series of the given joint member:
1) Importantly n = 1, n = 2, n = 3, there may be an uneven sequence of numbers: , , , having added up their terms, the row is removed
2) By doing it this way, we eliminate the series
3) Nadayuchi n values 1, 2, 3 and doctors, 1! = 1, 2! = 1×2, 3! = 1×2×3, remove the row
Butt 2. Know n the th member of the series behind the first numbers:
1) ; 2) ; 3) .
Example 3. Find the sum of members in the series:
1) It is known that the sums of members are low:
Let’s write down the sequence of the partial sums: …, , ….
The final member of this sequence is e. Otje,
The sequence of partial sums lies between the same. Well, the same amount of money converges.
2) This is an infinitely decreasing geometric progression, in which a 1 = , q = . The Vikorist formula can be eliminated. This means that the series converges and is equal to 1.
Similarity and divergence of number series. Sign of familiarityd'Alembert :
It is necessary to indicate that the boiling point is low. The series can only be agreed upon, what is your sleeping member? u n with uncirculated larger numbers n pragna to zero:
If so, then the row diverges - this is a sufficient sign of disparity in the row.
Sufficient signs of intimacy with positive members.
A sign of equalization of rows with positive members. The next series converges, so that its members do not outweigh the similar members of the other, which the series readily converges on; The next row diverges, as its members override the similar members of another row, which are likely to diverge.
When examining series for similarity and complexity, a geometric series is often used for this sign.
what to converge at |q|
Let's separate.
When the series are followed, the harmonic series is also adjusted
Yakshcho p= 1, then this series expands to a harmonic series, which is separate.
Yakshcho p< 1, то члены данного ряда больше соответствующих членов гармонического ряда и, значит, он расходится. При p> 1 possible geometric series, in any way | q| < 1; он является сходящимся. Итак, обобщенный гармонический ряд сходится при p> 1 i will pay for p£1.
Sign of d'Alembert. For order with positive members
(u n >0)
minds are converging, then the series converges l l > 1.
D'Alembert's sign does not give any indication, because l= 1. And here the research is low and other approaches are difficult to follow.
Significant rows.
The absolute intelligence of the rows:
Number series
u 1 + u 2 + u 3 + u n
It is called significant because the middle of its terms contains both positive and negative numbers.
A number series is called a sequential sign, as if two members stand in line, they create parallel signs. Let’s call this row a part of the familiar row.
A sign of danger for the rows that are being drawn.. As the members of the series, which are drawn, change monotonically with absolute value and the leading member u n pragna to zero at n® , then the series converges.
| A series is called absolutely similar if the series also converges. If a series converges absolutely, it is similar (in the primary sense). The turning point is not so. A series is called intellectually convergent because it tends to converge, but the series, folded from the moduli of its members, diverges. Example 4. Follow the row for shortness. |
| It is clear that Leibniz's sign is sufficient for the series that are being drawn. We remove the fragments. Well, this series is converging. Example 5. Follow the row for shortness. |
| Let’s try to establish the Leibniz sign: It can be seen that the modulus of the joint member does not vanish at n → ∞. Therefore this series diverges. Example 6. It means that these are a number of absolutely similar, mentally similar or dissimilar. |
| Using d'Alembert's sign for a series composed of modules of subordinate terms, we know that this series converges absolutely. |
Example 7. Observe for similarity (absolutely or intellectually) the row that the sign shows:
1) The members of this series monotonically decrease in absolute value. Then, according to Leibniz’s sign, the series converges. It is clear that it is absolutely intellectual to converge this series.
2) The terms of this series monotonically decrease in absolute value: , but
Functional rows:
The primary number series consists of numbers:
All members of the row are numbers.
The functional range consists of functions:
In the final member of the series, in addition to rich members, factorials, etc. immediately Enter the letter "IX". It looks, for example, like this: . Just like a number series, any functional series can be written in expanded view:
As you know, all members of the functional series are the same functions.
The most popular variety of the functional series is static row.
Step rows:
Next to the steps is called a series of minds
de numbers a 0, a 1, a 2, a n are called coefficients in a series, and a member a n x n- As a sleeping member in a row.
The areas of convergence of a static series are called without any significance x, in which series converge.
Number R is called the radius of proximity is low, because | x|
Stock 8. Denmark row
Follow his life at the points x= 1 i X= 3, x= -2.
When x = 1, the series is transformed into a number series
We can trace the passage of this row behind the D'Alembert sign. Maemo
Tobto. series converge.
At x = 3 the row is eliminated
To diverge, in order not to end, there is a need for a sign of convergence in the series
At x = -2 it is eliminated
The series, which, behind the sign of Leibniz, converges.
Ozhe, at the points x= 1 i X= -2. the series converges, but exactly x= 3 diverge.
Arrangement of elementary functions to the Maclaurin series:
Taylor's Bail for function f(x) is called statically close to view
Yakshcho, a = 0, then we reject the semiconductor of the Taylor series
which is called Maclaurin order.
The step row in the middle of its interval of convergence can be differentiated and integrated term by term as many times as possible, and the opposite rows follow the same interval of convergence as the outgoing row.
Two stacked series can be folded and multiplied term by term following the rules for folding and multiplying multiple terms. In this case, the run-up interval of the selected new row is avoided from the outer part of the run-out span of the output rows.
To expand the function to the Maclaurin series it is necessary:
1) calculate the values of the function and the last descendants of the point x = 0, then. . . .
8. Expand functions to the Maclaurin series.
Rows for teapots. Apply your decision
Having seen everyone, I’m on a different course! In this lesson, or rather, in a series of lessons, we will begin to walk in rows. The topic is not very complex, but for it to master it requires knowledge from the first course, in detail, it is necessary to understand, what is the boundary And you know the simplest boundaries. However, it’s okay, in due course I’ll give you special messages for the necessary lessons. For active readers, the topic of mathematical series, solutions, signs, and theorems may seem strange, or seem chimerical, mindless. In this case, there is no need to be very “fascinated”, we accept the facts as they are, and simply consider the typical, broad implications.
1) Rows for teapots, and for samovars there is a replacement :)
For advanced preparation on the topicє express course in pdf format, which will help you really “raise” your practice literally in a day.
Understanding the number line
Zagalom number series can be written like this: .
Here:
- Math sumi icon;
– member of the row(remember this simple term);
- Zminna is a “healer”. The record indicates that summation is carried out from 1 to “plus infinity”, then from now on, then, then, and so on - until infinity. The replacement of the changeable inode is changed either. The sum does not necessarily begin with one, sometimes it can begin with zero, with two, or with whatever natural number.
Apparently, before the change - “healer”, any row can be painted lit up: 
- And so on, ad infinitum.
Dodanki 
– tse NUMBERS what they are called members row. Like all the stench is unknown (More or equal to zero), then such a series is called positive number series.
Butt 1
This is already, before speaking, a “battle” task – it is practical to dosit and often need to write down a number of members in a row.
Start now, then:
Then then:
Then, then:
The process can be continued indefinitely, but in the end you need to write the first three terms of the series, so we write the following: 
Restore respect to the principle of humility numerical sequence,
which members are not assumed to have, but are seen as such.
Butt 2
Write down the first three terms in the row
This is a butt for independent decision-making, as a reminder of the lesson
However, for a row that seems folded at first glance, it is not difficult to paint it in a brightened view:
Butt 3
Write down the first three terms in the row
In reality, the legacy ends clearly: thoughts are presented to the sleeping member in a row first, then i. In the pouch: 
The truth is lost in such a look, you won't be able to feel the members of a row anymore, then don't hesitate dii: , , . Why? Confirmation at a glance 
It’s much easier and easier to check your accounts.
Sometimes the gates close in
Butt 4
There is no clear solution to the algorithm here, the regularity just needs to be learned.
In this section:
To reverse the cuts, the row can be painted backwards from the open view.
And the axle butt is slightly foldable for independent decisions:
Butt 5
Write down the sum of the bright-eyed member of the row 
Vikonati re-verify, again writing down the row in the flaming eyes
Number of number series
One of the key tasks is research is low in terms of profitability. In this case, two scenarios are possible:
1) Rowdiverge. This means that an endless sum of ancient inconsistencies: or the sums have burned can't sleep, yak, for example, near the row 
(Axis, before speech, and butt next to the negative members). A good example of a number series that diverges, focusing on the beginning of the lesson: 
. Here it is completely obvious that the skin of the forward member of the row is larger, lower than the front one, so 
And, well, the series diverges. An even more trivial butt: 
.
2) Rowconverge. This means that an infinite sum is as old as anyone end date: . Please: 
– this series converges and its sum is equal to zero. What is the best way to aim the buttstock? I'm falling endlessly geometric progression, familiar to us back in school: 
. The sum of the members of the endlessly declining geometric progression is calculated using the following formula: where is the first member of the progression, and is the basis, as, say, write as correct fractions In this category: , . In this order: 
The end number was removed, so the series converged, which needed to be completed.
However, in the most important cases know the amount in a row It’s not so simple, and therefore, in practice, in order to trace the viability of a number of special signs, they are developed theoretically.
There is a sign of low tenderness: a row sign is needed, alignment signs, d'Alembert's sign, Cauchy's sign, Leibniz sign and other signs. If I think I'm going to stagnate? These types of the male member are low, figuratively apparent - like the “filling” of the series. And very soon we will put everything in order.
! To further master the lesson it is necessary kindly understand What is the boundary and kindly reveal the insignificance of the view. To repeat or revise the material, go to the statistics Between. Apply your decision.
A row sign is required
If the series converges, its joint member will not be zero: .
It’s not true at the gate, then. If so, the series can either converge or diverge. And this sign is used for priming diversity row:
I am the most important member of the row no pragne scratch, then the series diverges
Or in short: if it is, then the series diverges. It’s okay, the situation is possible if there is no boundary between them, as, for example, boundaries. The axis is straight and wrapped around the separation of one row:)
But much more often, between the series, which diverge, there are ancient inconsistencies, in which the “dynamic” replacement of “x” appears in the framework of the “dynamic” changeable replacement of “x”. Let’s refresh our knowledge: boundaries with “ix” are called boundaries of functions, and boundaries with a variable “en” – boundaries of numerical sequences. The significance of the fact that the variable “en” takes on discrete (first) natural values is obvious: 1, 2, 3, etc. However, this fact has little bearing on the methods of unraveling between and methods of revealing inconsistencies.
Let us see that the row from the first butt diverges.
The last member of the series:
Visnovok: row diverge
The necessary sign is often found in real practical tasks:
Butt 6
We have a lot of articulations for the number and bannerman. The one who has respectfully read and comprehended the method of revealing the insignificance of statistics Between. Apply your decision, melodiously catching what if the senior level of number and standard equals also between the two end date .
Divide the number and the sign on 
Additional row diverge, because the necessary sign of proximity to the series is not indicated.
Butt 7
Follow the row for success
This is an example of independent decision. Outside the solution and the conclusion to the lesson
Well, if we are given ANY numerical series, in Pershu Chergu Let's check it out (thoughts in black): and why reduce your sleeping member to zero? In any case, we draw up a decision in accordance with exhibits No. 6, 7 and give evidence about those that diverge in a number of ways.
What types of rows have we looked at that obviously diverge? It immediately dawned on me that the rows diverge as much as possible. The row of butts No. 6, 7 also diverges: if there are many terms in the number book and the sign book, and the senior level of the number book is larger or equal to the senior level of the sign book. In all these cases, with the highest and highest quality of the butts, the necessary sign of viability is low.
Why is the sign called? necessary? Think more naturally: in order for the series to converge, necessary So that your sleeping member has jumped to zero. І everything would be good, but nothing else few. In other words, Since the second member of the series is zero, it DOES NOT MEAN that the series converges- You can, just as they converge, they diverge!
Get to know:
This row is called harmonious side by side. Be kind, remember! In the middle of the numerical series there is a prima ballerina. More precisely, a ballerina =)
It's easy to note that 
, ALE. Theoretically, mathematical analysis shows that the harmonic series diverges.
The same way to remember the concept of a regularized harmonic series:
1) This row diverge at . For example, the rows diverge, , .
2) This row converge at . For example, the rows , , . Once again I will say that in all practical tasks it is not at all important to us what the value of the bag is, for example, in a row, the very fact of his marriage is important.
Some elementary facts from the theory of series, as already established, and with any practical application, one can safely try, for example, to diverge the series or converge the series.
Look, the material, as you can see, is very similar to investigation of non-volatile integrals And for those who have mastered this topic, it will be easier. Well, for those who haven’t fought, it’s easier to win :)
So, why should we be afraid, what is the most important member of the row of pragne to zero? In such situations, for the best applications, it is necessary to use other methods, sufficient signs of divergence/disintegration:
Leveling signs for positive number series
I'll burn your respect, what is there only about positive number series (With unknown members).
There are two signs of equalization, one of which I simply call sign of leveling, other – leveling boundary sign.
Let's take a look now leveling sign, Or more precisely, I’m telling you this part:
Let's look at two positive number series i. As we know, what row – converge, and, starting from the current number, there is unevenness, then a series may converge.
In other words: The kinship of a series of larger members is matched by a kinship of a series of smaller members. In reality, inequality is often given meaning to everything:
Butt 8
Follow the row for success
First of all, let’s check(thoughts or in Chernetsi) Vikonanny: 
, Which means that “to survive with little blood” did not go away.
We look into the “pack” of the established harmonic series and, focusing on the senior level, we find a similar series: From the theory it is clear that we can converge.
For all natural numbers, obvious inequality is true:
and larger signs are represented by smaller fractions: 
, also, behind the sign of equalization there is a row converge at once from instructions.
If you have any doubts, then your nervousness can be written down in a report! We have written down the inconsistency for several numbers “en”:
Something like that
Something like that
Something like that
Something like that
….
and now it’s completely clear that uneasiness 
Vikonano for all natural numbers "En".
We analyze the sign of leveling and rising of the butt from an informal point of view. Still, why does the series converge? Why the axis? If the series converges, then it can be Kintsev sum: 
. And fragments of all members of the series less from the corresponding members of the series, then it is clear that the sum of the series cannot be greater than the number, and moreover, we cannot maintain inconsistency!
Similarly, you can increase the value of similar series: 
, , 
etc.
! Regain respect, that in all cases we have “pluses”. Obviously, even if there is one minus, it can seriously complicate the analysis leveling signs. For example, if a row is equalized in the same manner with a row that is about to go (to write a lot of uneasiness for the first members), then the mind will not be confused! Here you can twist and choose to align another similar row, for example, but not cause caution and other unnecessary difficulties. Therefore, to prove the viability of the series, it is much easier to vikory boundary sign of leveling(Div. offensive paragraph).
Butt 9
Follow the row for success
And in what way I tell you to look at it yourself another part of the signs of leveling:
As we know, what row – diverge, i, starting from this number (often from the very first), Vikonano nervousness, then a series may differ.
In other words: From the diversity among the smaller members there is also the diversity among the larger members.
What do you need to earn?
It is necessary to equalize the following series with the harmonious series that diverge. For a better understanding, consider a number of specific inequalities and compare the differences between them.
The solution and design of the lesson is similar to the lesson.
As it turned out, it is practically impossible to stagnate without just looking at the leveling sign. The right “working horse” of numerical series is boundary sign of leveling, and in terms of frequency, vicoristics can compete with any other D'Alembert's sign.
Limit sign of alignment of numerical positive series
Let's look at two positive number series i. As the relationship between the members of these rows is ancient ending number, subzero number: , then the rows converge or diverge at the same time.
When does the border sign of leveling become stagnant? The boundary sign of alignment is stagnant if the “stuffing” of the series is rich in members. Either there is one polynomial for the znamennik, or there are polynomials for the number manager and for the znamennik. Optionally, rich members can be under the roots.
Rozrobimos next to him, because the front level sign has stalled.
Butt 10
Follow the row for success
We compare the dates next to next, what to go to. Vikorist is bordering on the leveling sign. Apparently, it’s low - converge. How can we show what is ancient end, end-to-end, zero number, then it will be shown that the series can converge.
The end number has been removed from the zero, and the following series has been removed converge at once from instructions.
Why was the row itself chosen for the leveling? If we had chosen some other row from the “both” of the regular harmonic row, then we wouldn’t have succeeded end, end zero numbers (you can experiment).
Note: if we have a boundary sign of equalization, doesn't matter, in the order of folding the position of the sleeping members, in the examined butt the position could have been folded in a random way: - this would change the essence of the matter.