This article contains and structured the information necessary to solve almost any example on the topic of numerical ranks, from finding the sum of a number before studying it for convergence.

Overview of the article.

Let's start with the definitions of the alignment, alternating series and the concept of convergence. Next, consider standard ranks, such as a harmonic series, a generalized harmonic series, recall the formula for finding the amount of infinitely decreasing geometric progression. After that, we proceed to the properties of the converging series, we will focus on the necessary condition for the convergence of a series and voice sufficient signs of the convergence of a series. The theory will be diluted with the solution of the characteristic examples with detailed explanations.

Navigating page.

Basic definitions and concepts.

Let we have a numerical sequence where .

Let us give an example numerical sequence: .

Numerical row - this is the sum of the members of the numerical sequence of the form .

As an example of a numerical series, the sum of infinitely decreasing geometric progression with the denominator q \u003d -0.5 can be given: .

Call general member of the numerical series or K-th member of a series.

For the previous example, the overall member of the numerical series has the form.

Partial amount of numerical row - This is the sum of the species, where N is a certain natural number. They also refer to the N-oh partial sum of the numerical series.

For example, the fourth partial amount of the row there is .

Partial sums Forming an infinite sequence of partial sums of the numerical series.

For our series, N is a partial amount is located according to the formula of the first n member of the geometric progression , that is, we will have the following sequence of partial sums: .

Numeric row is called convergentIf there is a finite limit of the sequence of partial sums. If the limit of the sequence of partial sums of the numerical series does not exist or infinite, then the range is called drawn.

The sum of the concerned numerical series called the limit of the sequence of its partial sums, that is, .

In our example, therefore, a number It converges, and its amount is sixteen third: .

As an example, the diverging series can be given to the amount of geometric progression with the denominator greater than the unit: . N-Aya partial amount is determined by the expression and the limit of partial sums is infinite: .

Another example of a diverging numerical series is the amount of the species . In this case, N-aya partial amount can be calculated as. The limit of partial sums is infinite .

Amount of type called harmonic numerical next .

Amount of type where s is some valid number called generalized harmonious numeric nearby.

The specified definitions are sufficient to substantiate the following very often used statements, we recommend to remember them.

The harmonic row is divergent.

We prove the divergence of the harmonic series.

Suppose that a number converges. Then there is a finite limit of its partial sums. In this case, you can record and, which leads us to equality .

On the other hand,


Do not cause doubts the following inequalities. In this way, . The resulting inequality indicates us that equality It cannot be achieved, which contradicts our assumption about the convergence of the harmonic series.

Conclusion: the harmonic row diverges.

The sum of the geometric progression of the form with the denominator Q is a convergent numeric near, if, and divergent next to.

We prove it.

We know that the sum of the first N members of the geometric progression is by the formula .

With fair


What points to the convergence of a numerical series.

At q \u003d 1 have a numerical series . Its partial sums are like, and the limit of partial sums is infinite What points to the divergence of a number in this case.

If q \u003d -1, then the numeric number will take the form . Partial sums take a value for odd N, and for even n. From this we can conclude that the limit of partial sums does not exist and the row diverges.

With fair


What points to the divergence of a numerical series.

A generalized harmonic series converges at S\u003e 1 and diverges at.

Evidence.

For s \u003d 1, we obtain a harmonic row, and above we set it out.

For s Fair inequality for all natural k. Due to the divergence of the harmonic series, it can be argued that the sequence of its partial sums is unlimited (since there is no finite limit). Then the sequence of partial sums of the numerical series is the more unlimited (each member of this series is larger than the relevant member of the harmonic series), therefore, a generalized harmonic row is separated at s.

It remains to prove the convergence of the series at s\u003e 1.

We write a difference:



Obviously, then


Cut the resulting inequality for n \u003d 2, 4, 8, 16, ...



Using these results, the following actions can be carried out with the initial numeric number:



Expression It is an amount of geometric progression, the denominator of which is equal. Since we consider the case at s\u003e 1, then. therefore
. Thus, the sequence of partial sums of the generalized harmonic series at s\u003e 1 is increasing and at the same time limited from above the value, therefore, it has a limit that indicates the convergence of the row. Proof completed.

Numeric row is called aligningif all his members are positive, that is, .

Numeric row is called aligningIf the signs of his neighboring members are different. Singing numerical row can be written as or where .

Numeric row is called signedIf it contains an infinite set of both positive and negative members.

An alternating number of a numerical series is a special occasion of a alternate series.

Rows

are alignment, aligning and alternating, respectively.

For a banned series, there is the concept of absolute and conditional convergence.

absolutely convergentIf a row is converged from the absolute values \u200b\u200bof its members, that is, the alignmental numerical series converges.

For example, numeric rows and absolutely converge, because a number converges , which is the sum of infinitely decreasing geometric progression.

Announced row called conditionally convergentIf a row diverges, and the series converges.

As an example, a conventionally converging numerical series can be brought . Numerical row Compiled from the absolute values \u200b\u200bof the members of the initial series, consigned, as it is harmonic. At the same time, the initial number is convergent, which is easily installed with. Thus, a numeric alternating series Conditionally moving.

Properties of converging numeric rows.

Example.

Prove the convergence of a numerical series.

Decision.

We write a row in another form . The numerical series converges, since the generalized harmonic series is converging at s\u003e 1, and by virtue of the second property of converging numerical series will also converge with a numerical coefficient.

Example.

Little row converges.

Decision.

We convert the source row: . Thus, we received the sum of two numeric rows and, and each of them converges (see the previous example). Therefore, due to the third property of converging numeric rows, the initial series converges.

Example.

Prove the convergence of the numerical series And calculate its amount.

Decision.

This numerical series can be represented as a difference of two rows:

Each of these rows is the sum of infinitely decreasing geometric progression, therefore, is convergent. The third property of the converging series suggests that the initial numerical series converges. I calculate its amount.

The first member of the series is a unit, and the denominator of the corresponding geometric progression is 0.5, therefore, .

The first member of the row is 3, and the denominator of the corresponding infinitely decreasing geometric progression is 1/3, so .

We use the results obtained to find the amount of the original numerical series:

The necessary condition for the convergence of the series.

If a numeric series converges, the limit of its K-C member is zero :.

In the study of any numerical series for convergence, first of all, the implementation of the necessary convergence condition should be verified. Failure to comply with this condition indicates the divergence of the numerical series, that is, if, the row diverges.

On the other hand, it is necessary to understand that this condition is not sufficient. That is, the fulfillment of equality does not talk about the convergence of the numerical series. For example, for the harmonic series, the necessary condition of convergence is performed, and the row diverges.

Example.

Explore a numerical row on convergence.

Decision.

Check the necessary condition for the convergence of the numerical series:

Limit the n-th member of the numeric number is not zero, therefore, the row diverges.

Sufficient signs of convergence of the alignmental series.

When using sufficient signs for the study of numerical rows, convergence is constantly faced with, so we recommend contacting this section in difficulty.

Required and sufficient condition for the convergence of the alignmental numerical series.

For convergence of the alignmental numerical series It is necessary and enough for the sequence of its partial sums is limited.

Let's start with signs of comparison of rows. Their essence consists in comparing the studied numerical series with a number, convergence or divergence of which is known.

The first, second and third sign of comparison.

The first sign of comparing the ranks.

Let both be two alignmental numeric rows and an inequality for all k \u003d 1, 2, 3, ... then from the convergence of a number of convergence, and from the divergence of the series should be divergence.

The first sign of comparison is used very often and is a very powerful tool for the study of numerical rows for convergence. The main problem is the selection of a suitable row for comparison. A number for comparison usually (but not always) is chosen in such a way that the indicator of its K-C member is equal to the difference in the degree of the numerical degree and the denominator of the K without a member of the studied numerical series. For example, let the difference in the degree of the numerator and the denominator indicators are 2 - 3 \u003d -1, therefore, for comparison, we choose a row with a K-th member, that is, a harmonic row. Consider several examples.

Example.

Install the convergence or divergence of the row.

Decision.

Since the limit of the total member of the series is zero, the necessary condition for the convergence of the series is made.

It is easy to see that the inequality is fair for all natural K. We know that the harmonic row diverges, therefore, on the first sign of comparison, the initial series is also divergent.

Example.

Explore a numeric row on convergence.

Decision.

Prerequisite The convergence of the numerical row is performed, since . Obviously the fulfillment of inequality For any natural value k. A number converges, since a generalized harmonic series is converging for S\u003e 1. Thus, the first sign of comparison of the rows allows you to state the convergence of the original numerical series.

Example.

Determine the convergence or divergence of the numerical series.

Decision.

Therefore, the necessary condition for the convergence of the numerical series is fulfilled. What number to choose for comparison? It is suggested by a numerical series, and to determine s, carefully examine the numerical sequence. Members of the numerical sequence increase to infinity. Thus, starting from some number N (namely, with n \u003d 1619), members of this sequence will be greater than 2. Starting from this number N, the inequality is true. The numerical series converges in virtue of the first property of the converging series, as it turns out from the converging series, the discarding of the first N is 1 member. Thus, on the first sign of comparison, there are a number of convergence, and by virtue of the first property of the converging numeric rows it will also converge.

The second sign of comparison.

Let both be aligning numerical rows. If, then the convergence of the series follows convergence. If, from the divergence of the numerical series it follows the divergence.

Corollary.

If, then the convergence of one row follows the convergence of the other, and the divergence should be separated from the divergence.

We explore a row on convergence using the second sign of comparison. As a number, take a row. We will find the limit of the relationship of the K-sized members of the numerical series:

Thus, according to the second sign of comparison, the convergence of the numerical series follows the convergence of the original series.

Example.

Explore the convergence of a numerical row.

Decision.

Check the necessary condition for the convergence of the series . The condition is fulfilled. To apply the second sign of comparison, we take a harmonic row. We will find the limit of the relationship of K-s member:

Consequently, from the divergence of the harmonic series it follows the divergence of the initial series on the second sign of comparison.

For information, we give the third sign of the comparison of the ranks.

Third sign comparison.

Let both be aligning numerical rows. If a condition is satisfied from some number N, then convergence should be converged from the convergence of the series.

Sign of Dalamber.

Comment.

The sign of the Dalamber is valid if the limit is endless, that is, if , then a series converges if , then a row diverges.

If, the sign of the Dalamber does not provide information on convergence or divergence of the series and additional research is required.

Example.

Explore the numerical row on the convergence of the Dalamber.

Decision.

We check the fulfillment of the necessary condition for the numerical series convergence, the limit is calculated by:

The condition is fulfilled.

We use the sign of Dalamber:

Thus, the series converges.

Radical sign Cauchy.

Let be a sign-plating numeric row. If, the numeric series converges, if, the row diverges.

Comment.

The radical sign of Cauchy is true if the limit is infinite, that is, if , then a series converges if , then a row diverges.

If, the Cauchy radical sign does not provide information on convergence or divergence of a number and requires additional research.

It is usually easy to see cases when it is best to use a radical sign of Cauchy. A case is characteristic when a general member of a numerical series represents a significant expression. Consider several examples.

Example.

Explore the alignmental numerical row on convergence using a radical sign of Cauchy.

Decision.

. On the radical sign of Cauchy get .

Consequently, a series converges.

Example.

Whether a numeric row converges .

Decision.

We use the radical sign of Cauchy Therefore, the numeric series converges.

Integral sign Cauchy.

Let be a sign-plating numeric row. We will form a function of continuous argument y \u003d f (x), similar functions. Let the Y \u003d F (X) function positive, continuous and decreasing on the interval, where). Then in case of convergence incompatible integral The studied numerical series converges. If involved integral Different, the initial row is also diverted.

When checking the decrease in the function Y \u003d F (X), the theory can be useful on the interval from the section.

Example.

Explore a numerical row with positive member for convergence.

Decision.

The necessary condition for the convergence of the number is made, since . Consider a function. It is positive, continuous and descending on the interval. Continuity and the positivity of this function does not cause doubt, but on descending, we will stop more detail. Find a derivative:
. It is negative in the interval, therefore, the function decreases on this interval.

Sign of the convergence of Dalamber radical sign of Cauchy convergence integral sign of convergence of Cauchy

One of the common signs of comparison, which is found in practical examples, is a sign of Dalamber. Cauchy signs are less common, but also very popular. As always, I will try to set out the material simply, accessible and understandable. The topic is not the most difficult, and all the tasks to a certain extent are stencil.

Jean Lerone Daember is the famous French Mathematics of the 18th century. In general, Daember specialized in differential equations And on the basis of their research, he was engaged in ballistic, so that his majesty flew the cannonic kernels. At the same time, they did not forget about the numeric rods, not in vain, then the Sherngi Napoleonic troops so clearly converged and dispelled.

Before formulating the sign, consider an important question:
When do you need to apply a sign of the convergence of Dalamber?

First, let's start with the repetition. Recall cases when you need to apply the most chassis marketing sign of comparison. The limiting sign of comparison is applied when in the total member of the series:
1) There is a polynomial in the denominator.
2) The polynomials are in the numerator and in the denominator.
3) One or both polynomials can be under the root.

The main prerequisites for the use of the Dalamber feature are as follows:

1) In the overall member of the series ("filling" of a number) includes a number to a degree, for example, and so on. Moreover, it does not matter where this thing is located, in a numerator or in the denominator - it is important that it is present there.

2) The general member of the series includes factorial. With factorials we crossed the swords still at the lesson Number sequence and its limit. However, it will not prevent again to spread the touchscreen tablecloth:





…


…

! When using a sign of Dalamber, we just have to paint the factorial in detail. As in the previous paragraph, the factorial can be located at the top or bottom of the fraction.

3) If in the total member of the series there is a "chain of multipliers", for example,. This case is rare, but! In the study of such a series, it often makes a mistake - see example 6.

Together with degrees or (and) factorials in the filling of a number often meet polynomials, it does not change things - you need to use a sign of Dalamber.

In addition, in the total member of a number, a degree and factorial can meet simultaneously; can meet two factorial, two degrees, it is important to be there at least something Considered items - and this is just a prerequisite for the use of a sign of Dalamber.

Sign of Dalamber: Consider positive numerical series . If there is a limit of the subsequent member to the previous one: then:
a) with a number converge. In particular, the series converges at.
b) with a number diverge. In particular, the row diverges at.
c) for sign does not give a response. You need to use another feature. Most often, the unit is obtained in the case when the sign of the Dalamber is trying to apply where it is necessary to use a marking sign of comparison.

Who still has problems with limits or misunderstanding limits, consult a lesson Limits. Examples of solutions. Without an understanding of the limit and ability to disclose uncertainty further, unfortunately, not to move.

And now the long-awaited examples.

Example 1.

We see that in the general member of a number we have, and this is a faithful prerequisite that you need to use a sign of Dalamber. First, the complete solution and sample design, comments below.

We use a sign of Dalamber:

converges.

(1) Compile the ratio of the next member of the series to the previous one :. From the condition we see that the general member of the series. In order to get the following member of the series you need instead of substitute: .
(2) Get rid of four-story fractions. With a certain experiment, this step can be skipped.
(3) In the numerator reveal the brackets. In the denominator we take a four of the degree.
(4) Reducing on. Constant we take out the limit for the limit. In the numerator in brackets, we give such components.
(5) The uncertainty is eliminated by the standard method - the division of the numerator and the denominator on the "EN" to the high degree.
(6) We divide the numerals to the denominators, and we indicate the terms that seek to zero.
(7) We simplify the answer and make a note that with the conclusion that, on the basis of the Dalamber, the series under study converges.

In the considered example, in the total member of the series, we met a polynomial 2nd degree. What if there is a polynomial 3rd, 4th or higher? The fact is that if it is given a very high degree, then difficulties with the disclosure of the brackets will arise. In this case, a "turbo" solution can be used.

Example 2.

Take a similar range and exploring it for convergence.

First a complete solution, then comments:

We use a sign of Dalamber:

Thus, the series under study converge.

(1) Making a relation.
(2) Get rid of four-story fractions.
(3) Consider the expression in the numerator and expression in the denominator. We see that in the numerator you need to disclose brackets and erect into the fourth degree: what do I don't want to do at all. In addition, for those who are not familiar with Binom Newton, this task may not be impracticable. Let's analyze the eldest degrees: if we reveal the brackets at the top, we will get the oldest degree. Below we have the same elder degree:. By analogy with the previous example, it is obvious that with the depth division of the numerator and the denominator on our limit, one will receive a unit. Or, as mathematics say, polynomials and - one order of growth. Thus, it is quite possible to circle a relation to a simple pencil and immediately indicate that this thing is striving for a unit. Similarly, we paint with the second pair of polynomials: and, they too one order of growth, and their attitude seeks a unit.

In fact, such a "halica" could be checked in Example No. 1, but for a polynomial 2nd degree, such a solution is still somehow unsolonged. Personally, I do this: if there is a polynomial (or polynomials) of the first or second degree, I use the "long" way to solve the example 1. If the polynomial 3rd and more high degreesI use the "turbo" -metode according to the example of Example 2.

Example 3.

Examine a row on convergence

Complete solution and sample design at the end of class numerical sequences.
(4) Redfish everything that can be reduced.
(5) Constant we take out the limit for the limit. In the numerator reveal the brackets.
(6) Uncertainty Eliminate the standard method - the division of the numerator and the denominator on the "EN" to the high degree.

Example 5.

Examine a row on convergence

Complete solution and sample design at the end of the lesson

Example 6.

Examine a row on convergence

Sometimes there are rows, which in their stuffing contain "chain" of multipliers, this type of series has not yet been considered. How to explore a row with a "chain" of multipliers? Use a sign of Dalamber. But first to understand what is happening by the collapse of the row detail:

From the decomposition, we see that each next member of the series adds an additional factor in the denominator, therefore, if a common member of a series, then the next member of the series:
. Here often automatically make a mistake, formally by algorithm recording that

An exemplary sample solution may look like this:

We use a sign of Dalamber:

Thus, the series under study converges.

Before you start working with this topic, I advise you to view a section with terminology for numerical rows. It is especially worth paying attention to the concept of a common member of the series. If you have doubts about choosing a sign of convergence, I advise you to look the topic "Choosing a sign of convergence of numerical rows."

The sign of D "Alamber (or a sign of the Dalamber) is used to study the convergence of series, the general member of which is strictly greater than zero, i.e. $ u_n\u003e $ 0. Such rows are called strictly positive. In standard examples, the Alamber feature is used in the limit form.

Sign of d "Alamber (in limit form)

If $ \\ sum \\ limits_ (n \u003d 1) ^ (\\ infty) u_n $ is strictly positive and $$ \\ Lim_ (N \\ To \\ infty) \\ FRAC (U_ (N + 1)) (U_N) \u003d L, $ $ then with $ l<1$ ряд сходится, а при $L>$ 1 (and at $ l \u003d \\ infty $) a row diverges.

The wording is quite simple, but the next question remains open: what will happen if $ L \u003d 1 $? Answer to this question is not able to give a sign. If $ L \u003d $ 1, then a row can both converge and disperse.

Most often in standard examples, the Alamber feature is applied if there is a polynomial from $ n $ in the expression of a common member of the series (the polynomial may be under root) and the degree of type $ a ^ n $ or $ n! $. For example, $ u_n \u003d \\ FRAC (5 ^ n \\ CDOT (3N + 7)) (2N ^ 3-1) $ (see example No. 1) or $ U_n \u003d \\ FRAC (\\ SQRT (4N + 5)) ((3N-2)$ (см. пример №2). Вообще, для стандартного примера наличие $n!$ - это своеобразная "!} business card"Sign of D" Alamber.

What denotes the expression "n!"? Show \\ Hide

Recording "N!" (read "en factorial") means the work of all natural numbers From 1 to N, i.e.

$$ n! \u003d 1 \\ CDOT2 \\ CDOT 3 \\ CDOT \\ LDOTS \\ CDOT N $$

By definition, it is assumed that $ 0! \u003d 1! \u003d $ 1. For example, find 5!:

$$ 5! \u003d 1 \\ CDOT 2 \\ CDOT 3 \\ CDOT 4 \\ CDOT 5 \u003d 120. $$.

In addition, the sign of D "Alamber is used to determine the convergence of a series, the general member of which contains a product of such a structure: $ U_N \u003d \\ FRAC (3 \\ CDOT 5 \\ CDOT 7 \\ CDOT \\ LDOTS \\ CDOT (2N + 1)) (2 \\ Example №1

Explore a number $ \\ sum \\ limits_ (n \u003d 1) ^ (\\ infty) \\ FRAC (5 ^ n \\ Cdot (3N + 7)) (2n ^ 3-1) $ for convergence.

Since the lower summation limit is 1, then the total member of the row is recorded under the sum of the amount: $ U_n \u003d \\ FRAC (5 ^ n \\ Cdot (3N + 7)) (2N ^ 3-1) $. Since with $ n≥ 1 $ we have $ 3N + 7\u003e 0 $, $ 5 ^ n\u003e 0 $ and $ 2n ^ 3-1\u003e 0 $, then $ u_n\u003e 0 $. Consequently, our row is strictly positive.

$$ 5 \\ CDOT \\ LIM_ (n \\ to \\ infty) \\ FRAC ((3n + 10) \\ left (2n ^ 3-1 \\ Right)) (\\ left (2 (n + 1) ^ 3-1 \\ RIGHT ) (3n + 7)) \u003d \\ left | \\ FRAC (\\ INFTY) (\\ infty) \\ Right | \u003d 5 \\ Cdot \\ Lim_ (n \\ to \\ infty) \\ FRAC (\\ FRAC ((3N + 10) \\ Left (2n ^ 3-1 \\ Right)) (n ^ 4)) (\\ FRAC (\\ left (2 (n + 1) ^ 3-1 \\ RIGHT) (3N + 7)) (n ^ 4)) \u003d 5 \\ Cdot \\ Lim_ (n \\ to \\ infty) \\ FRAC (\\ FRAC (3N + 10) (N) \\ CDOT \\ FRAC (2N ^ 3-1) (N ^ 3)) (\\ FRAC (\\ Left (2 ( N + 1) ^ 3-1 \\ RIGHT)) (n ^ 3) \\ CDOT \\ FRAC (3N + 7) (N)) \u003d \\\\ \u003d 5 \\ Cdot \\ Lim_ (N \\ To \\ infty) \\ FRAC (\\ \\ Right)) (\\ left (2 \\ left (\\ FRAC (N) (N) + \\ FRAC (1) (N) \\ Right) ^ 3- \\ FRAC (1) (N ^ 3) \\ Right) \\ CDOT \\ left (\\ FRAC (3N) (N) + \\ FRAC (7) (N) \\ RIGHT)) \u003d 5 \\ CDOT \\ LIM_ (N \\ To \\ infty) \\ FRAC (\\ left (3+ \\ FRAC (10) (n) \\ Right) \\ Cdot \\ left (2- \\ FRAC (1) (N ^ 3) \\ Right)) (\\ left (2 \\ left (1+ \\ FRAC (1) (N) \\ RIGHT) ^ 3 - \\ FRAC (1) (N ^ 3) \\ Right) \\ Cdot \\ Left (3+ \\ FRAC (7) (N) \\ Right)) \u003d 5 \\ CDOT \\ FRAC (3 \\ CDOT 2) (2 \\ CDOT 3 ) \u003d 5. $$.

Since $ \\ lim_ (n \\ to \\ infty) \\ FRAC (U_ (N + 1)) (U_N) \u003d 5\u003e $ 1, then according to the specified row diverges.

Honestly, a sign of D "Alamber is not the only option in this situation. You can use, for example, a radical sign of Cauchy. However, the use of a radical sign of Cauchy will require knowledge (or evidence) of additional formulas. Therefore, the use of a sign D" Alamber in this situation is more convenient.

Answer: A row diverges.

Example number 2.

Explore a number $ \\ sum \\ limits_ (n \u003d 1) ^ (\\ infty) \\ FRAC (\\ SQRT (4N + 5)) ((3n-2)$ на сходимость.!}

Since the lower summation limit is 1, then the total member of the row is recorded under the sum of the amount: $ U_n \u003d \\ FRAC (\\ SQRT (4N + 5)) ((3N-2)$. Заданный ряд является строго положительным, т.е. $u_n>0$.!}

The general member of the row contains a polynomial under the root, i.e. $ \\ SQRT (4N + 5) $, and factorial $ (3N-2)! $. The presence of a factorial in a standard example is an almost 100% guarantee of the applying of the D "Alamber.

To apply this feature, we will have to find the rating limit of $ \\ FRAC (U_ (n + 1)) (u_n) $. To record $ U_ (n + 1) $, you need in a formula $ U_n \u003d \\ FRAC (\\ SQRT (4N + 5)) ((3N-2)$ вместо $n$ подставить $n+1$:!}

$$ U_ (n + 1) \u003d \\ FRAC (\\ SQRT (4 (N + 1) +5)) ((3 (n + 1) -2)=\frac{\sqrt{4n+9}}{(3n+1)!}. $$ !}

Since $ (3n + 1)! \u003d (3N-2)! \\ CDOT (3N-1) \\ CDOT 3N \\ CDOT (3N + 1) $, then the formula for $ U_ (n + 1) $ can be recorded In a variety:

$$ U_ (n + 1) \u003d \\ FRAC (\\ SQRT (4N + 9)) ((3n + 1)=\frac{\sqrt{4n+9}}{(3n-2)!\cdot (3n-1)\cdot 3n\cdot(3n+1)}. $$ !}

This entry is convenient for a further solution when we have to shorten the fraction under the limit. If equality with factorials requires explanations, please reveal the note below.

How we got the equality $ (3n + 1)! \u003d (3N-2)! \\ CDot (3N-1) \\ CDOT 3N \\ CDOT (3N + 1) $? Show \\ Hide

Recording $ (3N + 1)! $ Means the product of all natural numbers from 1 to $ 3N + $ 1. Those. This expression can be written like this:

$$ (3N + 1)! \u003d 1 \\ CDOT 2 \\ CDOT \\ LDOTS \\ CDOT (3N + 1). $$.

Directly in front of the number of $ 3n + 1 $ there is a number, per unit smaller, i.e. The number is $ 3N + 1-1 \u003d 3n $. And immediately before the number of $ 3n $ costs the number of $ 3N-1 $. Well, just before the number of $ 3n-1 $ we have a number $ 3N-1-1 \u003d 3N-2 $. We rewrite the formula for $ (3n + 1)! $:

$$ (3N + 1)! \u003d 1 \\ CDOT2 \\ CDOT \\ LDOTS \\ CDOT (3N-2) \\ CDOT (3N-1) \\ CDOT 3N \\ CDOT (3N + 1) $$

What is a product of $ 1 \\ CDOT2 \\ CDOT \\ LDOTS \\ CDOT (3N-2) $? This product is $ (3N-2)! $. Consequently, the expression for $ (3n + 1)! $ Can rewrite in this form:

$$ (3n + 1)! \u003d (3N-2)! \\ CDOT (3N-1) \\ CDOT 3N \\ CDOT (3N + 1) $$

This entry is convenient for a further solution when we have to shorten the fraction under the limit.

Calculate the value of $ \\ lim_ (n \\ to \\ infty) \\ FRAC (U_ (n + 1)) (u_n) $:

$$ \\ LIM_ (n \\ to \\ infty) \\ FRAC (U_ (N + 1)) (U_N) \u003d \\ Lim_ (n \\ to \\ infty) \\ FRAC (\\ FRAC (\\ SQRT (4N + 9)) (( 3N-2)! \\ CDOT (3N-1) \\ CDOT 3N \\ CDOT (3N + 1))) (\\ FRAC (\\ SQRT (4N + 5)) ((3N-2)}= \lim_{n\to\infty}\left(\frac{\sqrt{4n+9}}{\sqrt{4n+5}}\cdot\frac{(3n-2)!}{(3n-2)!\cdot (3n-1)\cdot 3n\cdot(3n+1)}\right)=\\ =\lim_{n\to\infty}\frac{\sqrt{4n+9}}{\sqrt{4n+5}}\cdot\lim_{n\to\infty}\frac{1}{(3n-1)\cdot 3n\cdot(3n+1)}= \lim_{n\to\infty}\frac{\sqrt{4+\frac{9}{n}}}{\sqrt{4+\frac{5}{n}}}\cdot\lim_{n\to\infty}\frac{1}{(3n-1)\cdot 3n\cdot(3n+1)}=1\cdot 0=0. $$ !}

Since $ \\ lim_ (n \\ to \\ infty) \\ FRAC (U_ (n + 1)) (u_n) \u003d 0<1$, то согласно

Signs of convergence of rows.
Sign of Dalamber. Signs of Cauchy

Work, work - and understanding will come later
J.L. Daember

I congratulate everyone on the beginning of the school year! Today, on September 1, and I decided to introduce readers in honor of the holiday with the fact that you were looking forward to looking forward and eager to find out - signs of convergence of numerical positive rows. The first of September and my congratulations are always relevant, nothing terrible, if in fact, the summer outside the window, you now relieve the exam for the third time, if you went to this page!

For those who are just starting to study the ranks, I recommend to get acquainted with the article Numeric rows for teapots. Actually, this cart is a continuation of the banquet. So today at the lesson we will consider examples and decisions on the themes:

One of the common signs of comparison, which is found in practical examples, is a sign of Dalamber. Cauchy signs are less common, but also very popular. As always, I will try to set out the material simply, accessible and understandable. The topic is not the most difficult, and all the tasks to a certain extent are stencil.

Sign of convergence of Dalamber

Jean Lerone Daember is the famous French Mathematics of the 18th century. In general, Daember specialized in differential equations and on the basis of his research was engaged in ballistic, so that his majesty flew the cannonic kernels. At the same time, they did not forget about the numeric rods, not in vain, then the Sherngi Napoleonic troops so clearly converged and dispelled.

Before formulating the sign, consider an important question:
When do you need to apply a sign of the convergence of Dalamber?

First, let's start with the repetition. Recall cases when you need to apply the most chassis marketing sign of comparison. The limiting sign of comparison is applied when in the total member of the series:

1) There is a polynomial in the denominator.
2) The polynomials are in the numerator and in the denominator.
3) One or both polynomials can be under the root.
4) polynomials and roots, of course, maybe more.

The main prerequisites for the use of the Dalamber feature are as follows:

1) In the overall member of the series ("stuffing" of a number) includes a number to a degree, for example,, and so on. Moreover, it does not matter where this thing is located, in a numerator or in the denominator - it is important that it is present there.

2) The general member of the series includes factorial. With factoria, we crossed the swords even at the lesson the number sequence and its limit. However, it will not prevent again to spread the touchscreen tablecloth:





…


…

! When using a sign of Dalamber, we just have to paint the factorial in detail. As in the previous paragraph, the factorial can be located at the top or bottom of the fraction.

3) if there is a "chain of multipliers" in the total member of the series, for example, . This case is rare, but! In the study of such a series, it often makes a mistake - see example 6.

Together with degrees or (and) factorials in the filling of a number often meet polynomials, it does not change things - you need to use a sign of Dalamber.

In addition, in the total member of a number, a degree and factorial can meet simultaneously; can meet two factorial, two degrees, it is important to be there at least something From the considered items - and this is just a prerequisite for the use of the sign of the Dalamber.

Sign of Dalamber: Consider positive numerical series . If there is a limit of the subsequent member to the previous one: then:
a) with a number converge
b) with a number diverge
c) for sign does not give a response. You need to use another feature. Most often, the unit is obtained in the case when the sign of the Dalamber is trying to apply where it is necessary to use a marking sign of comparison.

Who still has problems with limits or misunderstanding limits, consult a lesson Limits. Examples of solutions. Without an understanding of the limit and ability to disclose uncertainty further, unfortunately, not to move.

And now the long-awaited examples.

Example 1.

We see that in the general member of a number we have, and this is a faithful prerequisite that you need to use a sign of Dalamber. First, the complete solution and sample design, comments below.

We use a sign of Dalamber:


converges.
(1) Compile the ratio of the next member of the series to the previous one :. From the condition we see that the general member of the series. In order to get the next member of the series you need Instead of substitute: .
(2) Get rid of four-story fractions. With a certain experiment, this step can be skipped.
(3) In the numerator reveal the brackets. In the denominator we take a four of the degree.
(4) Reducing on. Constant we take out the limit for the limit. In the numerator in brackets, we give such components.
(5) The uncertainty is eliminated by the standard method - the division of the numerator and the denominator on the "EN" to the high degree.
(6) We divide the numerals to the denominators, and we indicate the terms that seek to zero.
(7) We simplify the answer and make a note that with the conclusion that, on the basis of the Dalamber, the series under study converges.

In the considered example, in the total member of a number, we met a polynomial 2nd degree. What if there is a polynomial 3rd, 4th or higher? The fact is that if it is given a very high degree, then difficulties with the disclosure of the brackets will arise. In this case, a "turbo" solution can be used.

Example 2.

Take a similar range and exploring it for convergence.

First a complete solution, then comments:

We use a sign of Dalamber:


Thus, the series under study converge.

(1) Making a relation.

(3) Consider the expression In the numerator and expression in the denominator. We see that in the numerator you need to disclose brackets and erect into the fourth degree: what do I don't want to do at all. And for those who are not familiar with Binom Newton, this task will be even more difficult. Let's analyze older degrees: if we reveal brackets at the top , I get the older degree. At the bottom we have the same senior degree :. By analogy with the previous example, it is obvious that with the depth division of the numerator and the denominator on our limit, one will receive a unit. Or, as mathematics say, polynomials and - one order of growth. Thus, it is quite possible to circulate Simple pencil and immediately indicate that this thing is striving for a unit. Similarly, we paint with the second pair of polynomials: and, they too one order of growth, and their attitude seeks a unit.

In fact, such a "hacktur" could be checked in example 1, but for the polynomial of the 2nd degree, such a solution looks still somehow unsolving. Personally, I do this: if there is a polynomial (or polynomial) of the first or second degree, I use the "long" method of solving the example 1. If the polynomial of 3rd and higher degrees comes across, I use the "turbo" method according to the example of Example 2.

Example 3.

Examine a row on convergence

Consider typical examples with factorials:

Example 4.

Examine a row on convergence

The general member of the series includes a degree and factorial. It is clear how the day that it is necessary to use a sign of Dalamber. We decide.

Thus, the series under study diverge.
(1) Making a relation. We repeat again. By condition, the general member of the series: . In order to get the following member of the row instead you need to substitute, in this way: .
(2) Get rid of four-story fractions.
(3) press off the seven. Factorials describe in detail. How to do it - see the beginning of the lesson or article about numerical sequences.
(4) Redfish everything that can be reduced.
(5) Constant we take out the limit for the limit. In the numerator reveal the brackets.
(6) Uncertainty Eliminate the standard method - the division of the numerator and the denominator on the "EN" to the high degree.

Example 5.

Examine a row on convergence

Complete solution and sample design at the end of the lesson

Example 6.

Examine a row on convergence

Sometimes there are rows, which in their stuffing contain "chain" of multipliers, this type of series has not yet been considered. How to explore a row with a "chain" of multipliers? Use a sign of Dalamber. But first to understand what is happening by the collapse of the row detail:

From the decomposition, we see that each next member of the series adds an additional multiplier in the denominator, so if the general member of the series , then the next member of the series:
. Here often automatically make a mistake, formally by algorithm recording that

An exemplary sample solution may look like this:

We use a sign of Dalamber:

Thus, the series under study converges.

Cauchy radical sign

Augusten Louis Cauchy is an even more famous French mathematician. Cauchy's biography You can tell any student of a technical specialty. In the most picturesque paints. It is not by chance that this surname is carved on the first floor of the Eiffel Tower.

The sign of convergence of Cauchy for positive numeric rows is something similar to the just considered sign of Dalamber.

Cauchy radical sign:Consider positive numerical series . If there is a limit: then:
a) with a number converge. In particular, the series converges at.
b) with a number diverge. In particular, the row diverges at.
c) for sign does not give a response. You need to use another feature. It is interesting to note that if the sign of Cauchy does not give us a response to the question of the convergence of a number, then the sign of Dalamber will also not give an answer. But if the sign of Dalamber does not give a response, the sign of Cauch may well "work". That is, the sign of Cauchy is in this sense a stronger sign.

When should I use the Kauchi radical sign? The radical sign of Cauchy usually uses in cases where the root "good" is extracted from the total member of the series. As a rule, this pepper is in degree, which depends on . There are still exotic cases, but they will not score head.

Example 7.

Examine a row on convergence

We see that the fraction is completely under degree, depending on the "En", and therefore, it is necessary to use a radical sign of Cauchy:


Thus, the series under study diverge.

(1) We draw up a common member of a series of root.

(2) rewrite the same thing, only without a root, using the property of degrees.
(3) In the indicator, the numerator on the denominator, indicating that
(4) As a result, we turned out uncertainty. Here it was possible to go a long way: to build into a cube, build into a cube, then divide the numerator and denominator on "En" in Cuba. But in this case there is a more efficient solution: this reception can be used directly under the degree of constant. To eliminate uncertainty divide the numerator and denominator on (the older degree of polynomials).

(5) We carry out the soil division, and indicate the terms that seek to zero.
(6) I bring the answer to mind, we note that we conclude that a row diverges.

But a simpler example for an independent solution:

Example 8.

Examine a row on convergence

And a couple of typical examples.

Complete solution and sample design at the end of the lesson

Example 9.

Examine a row on convergence
We use the Kauchi radical sign:


Thus, the series under study converge.

(1) We put a general member of a row.

(2) rewrite the same thing, but already without a root, while revealing brackets using the formula of abbreviated multiplication: .
(3) In the indicator, the numerator on the denominator is renovated and indicate that.
(4) The uncertainty of the species is obtained, and here you can also perform division directly under the degree. But with one condition: The coefficients for senior degrees of polynomials must be different. We have different (5 and 6), and therefore it is possible (and necessary) to divide both floors on. If these coefficients the same, for example (1 and 1):, then such a focus does not pass and you need to use the second wonderful limit. If you remember, these subtleties were considered in the last paragraph of the article Methods for solving limits.

(5) actually perform the soil division and indicate which terms we tend to zero.
(6) Uncertainty Eliminated, we have the simplest limit :. Why in infinitely big degree tends to zero? Because the foundation of the degree satisfies inequality. If anyone has doubts about the justice of the limit , I'm not lazy, I will take a calculator in my hands:
If, then
If, then
If, then
If, then
If, then
… etc. To infinity - that is, in the limit:

Bily infinitely decreasing geometric progression on the fingers \u003d)
! Never use this technique as proof! For if something is obvious, it does not mean that it is correct.

(7) We indicate that we conclude that the series converges.

Example 10.

Examine a row on convergence

This is an example for an independent solution.

Sometimes a provocative example is proposed to solve, for example:. Here in an indicator no "En", only constant. Here you need to build a numerator and denominator to the square (the polynomials are obtained), and then adhere to the algorithm from the article Rows for teapots. In such an example, it should be done either the necessary sign of the convergence of a number or a limit sign of comparison.

Integral sign Cauchy

Or just an integral sign. Disappointing those who poorly learned the first course material. In order to apply the integral feature of Cauchy, it is necessary to more or less confidently be able to find derivatives, integrals, as well as having the calculation skill incompatible integral First kind.

In textbooks on mathematical analysis integral sign Cauchy Dan mathematically strictly, but too shaky, so I formulate a sign not too strictly, but it is clear:

Consider positive numerical series . If there is a immutable integral, then a series converges or diverge with this integral.

And immediately examples for explanation:

Example 11.

Examine a row on convergence

Almost classic. Natural logarithm and some kind of bjaka.

The main prerequisite for using the integral sign of Cauchy It is the fact that in the total member of a number contains multipliers similar to some function and its derivative. From the topic