Study of alternating series. alternating series

Definition 6.1 A number series containing an infinite set of positive and an infinite set of negative terms is called alternating. A special case of an alternating series is an alternating series, that is, a series in which successive terms have opposite signs.

Leibniz sign

For those alternating nearby, the Leibniz sufficient convergence test applies.

Let (an) be a number sequence such that

1. an+1< an ;

Then the alternating series and converge.

Absolute and conditional convergence

Definition 6.2 A series is said to be absolutely convergent if the series also converges. If a series converges absolutely, then it is convergent (in the usual sense). The converse is not true.

A series is said to be conditionally convergent if it itself converges and the series composed of the modules of its members diverges.

Let us apply the Leibniz sufficient test for alternating series. We get

insofar as. Therefore, this series converges.

Examine the series for convergence.

Let's try to apply the Leibniz sign:

It can be seen that the modulus of the general term does not tend to zero for n > ?. So this series diverges

Applying the d'Alembert criterion to a series composed of the modules of the corresponding terms, we find

Therefore, this series converges absolutely.

Determine if the series is absolutely convergent, conditionally convergent, or divergent?

First, we use the Leibniz sign and find the limit. Let's calculate this limit according to L'Hopital's rule:

Thus, the original series diverges.

Investigate for convergence series

The common term of this series is equal to. Let's apply the d'Alembert test to a series composed of modules:

Consequently. the original series converges absolutely.

Investigate whether the series is absolutely convergent, conditionally convergent, or divergent?

Applying the Leibniz test, we see that the series is convergent:

Let us now consider the convergence of a series composed of the modules of the corresponding terms. Using the integral criterion of convergence, we obtain

Therefore, the original series converges conditionally.

Determine if the series is absolutely convergent, conditionally convergent, or divergent?

First we apply the Leibniz test:

Therefore, this series converges. Let us find out whether this convergence is absolute or conditional. Let's use the limit test of comparison and compare the corresponding series of modules with the divergent harmonic series:

Since the series composed of modules diverges, the original alternating series is conditionally convergent.

1. Series with positive terms. Signs of convergence

It is very difficult to determine the convergence of the series (1.1) and find its sum in the case of convergence directly by Definition 1.1 as the limit of a sequence of partial sums. Therefore, there are sufficient signs for determining whether the series converges or diverges. In the case of its convergence, the approximate value of its sum with any degree of accuracy can be the sum of the corresponding number of the first n terms of the series.

Here we will consider series (1.1) with positive (nonnegative) terms, i.e., series for which such series will be called positive series.

Theorem 3.1. (comparison sign)

Let there be two positive rows

and the conditions are met for all n=1,2,…

Then: 1) the convergence of the series (3.2) implies the convergence of the series (3.1);

2) the divergence of the series (3.1) implies the divergence of the series (3.2).

Proof. 1. Let the series (3.2) converge and its sum be equal to B. The sequence of partial sums of the series (3.1) is non-decreasing and bounded from above by the number B, i.e.

Then, due to the properties of such sequences, it follows that it has a finite limit, i.e., the series (3.1) converges.

2. Let series (3.1) diverge. Then, if the series (3.2) converges, then, by virtue of item 1 proved above, the original series would also converge, which contradicts our condition. Hence series (3.2) also diverges.

This feature is conveniently applied to the determination of the convergence of series by comparing them with series whose convergence is already known.

Example 3.1. Investigate for convergence series

The terms of the series are positive and less than the corresponding terms of the convergent series of a geometric progression

because, n=1,2,…

Therefore, by comparison, the original series also converges.

Example 3.2. Investigate for convergence series

The terms of this series are positive and greater than the corresponding terms of the divergent harmonic series

Therefore, according to the criterion of comparison, the original series diverges.

Theorem 3.2. (The limiting sign of d'Alembert).

Then: 1) for q< 1 ряд (1.1) сходится;

2) for q > 1 series (1.1) diverges;

Remark: Series (1.1) will also diverge when

Example 3.3. Investigate for convergence series

Let's apply the d'Alembert limit test.

In our case.

Example 3.4. Investigate for convergence series

Therefore, the original series converges.

Example 3.5. Investigate for convergence series

Let's apply the d'Alembert limit test:

Therefore, the original series diverges.

Comment. The application of the d'Alembert limit test to a harmonic series does not give an answer about the convergence of this series, because for this series

Theorem 3.3. (The Cauchy Limit Cauchy Augustin Louis (1789 - 1857), French mathematician.).

Let the terms of the positive series (1.1) be such that there exists a limit

Then: 1) for q< 1 ряд (1.1) сходится;

2) for q > 1 series (1.1) diverges;
3) for q = 1, nothing can be said about the convergence of the series (1.1), additional studies are needed.

Example 3.6. Investigate for convergence series

We apply the Cauchy limit test:

Therefore, the original series converges.

Theorem 3.4. (Integral Cauchy test).

Let the function f(x) be a continuous non-negative non-increasing function on the interval

Then the series and the improper integral converge or diverge simultaneously.

Example 3.7. Examine the harmonic series for convergence

We apply the Cauchy integral test.

In our case, the function satisfies the condition of Theorem 3.4. We investigate the convergence of the improper integral

The improper integral diverges, therefore, the original harmonic series also diverges.

Example 3.8. Investigate the generalized harmonic series for convergence

The function satisfies the condition of Theorem 3.4.

We investigate the convergence of the improper integral

Consider the following cases:

1) let Then the generalized harmonic series is a harmonic series that diverges, as shown in Example 3.7.
2) let Then

The improper integral diverges, and hence the series diverges;

3) let Then

The improper integral converges, and hence the series converges.

Finally we have

Remarks. 1. The generalized harmonic series will diverge at, since in this case the necessary convergence criterion is not satisfied: the common term of the series does not tend to zero.

2. The generalized harmonic series is convenient to use when applying the comparison criterion.

Example 3.9. Investigate for convergence series

The terms of the series are positive and less than the corresponding terms of the convergent generalized harmonic series

because and parameter

Therefore, the original series converges (on the basis of comparison).

Let us turn to the consideration of series, the terms of which can be both positive and negative.

Alternating series are series whose terms are alternately positive and negative. . Most often, alternating series are considered, in which the terms alternate through one: each positive is followed by a negative, each negative is followed by a positive. But there are alternating rows in which the members alternate after two, three, and so on.

Consider an example of an alternating series, the beginning of which looks like this:

3 − 4 + 5 − 6 + 7 − 8 + ...

and immediately the general rules for writing alternating series.

As in the case of any series, to continue this series, you need to specify a function that determines the common term of the series. In our case, this n + 2 .

And how to set the alternation of signs of the members of the series? Multiplying the function by minus one to some degree. In what degree? We emphasize right away that not any degree provides alternation of signs at the terms of the series.

Let's say we want the first term of an alternating series to be positive, as is the case in the above example. Then minus one must be in the power n− 1 . Start substituting numbers starting from one into this expression and you will get as an exponent at minus one, then an even, then an odd number. This is the necessary condition for the alternation of signs! We get the same result when n+ 1 . If we want the first term of the alternating series to be negative, then we can specify this series by multiplying the common term function by one to the power n. We get an even number, then an odd number, and so on. As you can see, the already described condition for the alternation of signs is fulfilled.

Thus, we can write the above alternating series in general form:

For alternating signs of a term of a series, the power minus one can be the sum n and any positive or negative, even or odd number. The same applies to 3 n , 5n, ... That is, the alternation of the signs of the members of the alternating series provides the degree at minus one in the form of a sum n multiplied by any odd number and any number.

What degrees at minus one do not provide alternation of signs of the members of the series? Those that are present in the form n multiplied by any even number, to which is added any number, including zero, even or odd. Examples of indicators of such degrees: 2 n , 2n + 1 , 2n − 1 , 2n + 3 , 4n+ 3 ... In the case of such degrees, depending on the number with which "en" is added, multiplied by an even number, either only even or only odd numbers are obtained, which, as we have already found out, does not give alternation of signs of the members of the series .

Alternating series - a special case alternating series . Alternating series are series with members of arbitrary signs , that is, those that can be positive and negative in any order. An example of an alternating series:

3 + 4 + 5 + 6 − 7 + 8 − ...

Next, consider the convergence criteria for alternating and alternating series. The conditional convergence of alternating series can be established using the Leibniz test. And for a wider range of series - alternating (including alternating) - there is a sign of absolute convergence.

Convergence of alternating series. Leibniz sign

For alternating series, the following test of convergence takes place - the Leibniz test.

Theorem (Leibniz test). The series converges, and its sum does not exceed the first term, if the following two conditions are simultaneously satisfied:

the absolute values of the members of the alternating series decrease: u1 > u 2 > u 3 > ... > u n > ...;
limit of its common term with unlimited increase n equals zero.

Consequence. If for the sum of an alternating series we take the sum of its n terms, then the error allowed in this case will not exceed the absolute value of the first discarded term.

Example 1 Investigate the convergence of a series

Solution. This is an alternating row. The absolute values of its members decrease:

and the limit of the common term

equals zero:

Both conditions of the Leibniz test are satisfied, so the series converges.

Example 2 Investigate the convergence of a series

Solution. This is an alternating row. Let's first prove that:

, .

If N= 1 , then for all n > N inequality 12 n − 7 > n. In turn, for each n. Therefore, that is, the terms of the series decrease in absolute value. Let us find the limit of the common term of the series (using L'Hopital's rule):

The limit of the common term is zero. Both conditions of the Leibniz criterion are met, so the answer to the question of convergence is positive.

Example 3 Investigate the convergence of a series

Solution. An alternating series is given. Let us find out whether the first condition of the Leibniz sign is satisfied, that is, the requirement . For the requirement to be met, it is necessary that

We made sure that the requirement is met for all n > 0 . The first Leibniz test is satisfied. Find the limit of the common term of the series:

The limit is not zero. Thus, the second condition of the Leibniz test is not satisfied, so convergence is out of the question.

Example 4 Investigate the convergence of a series

Solution. In this series, two negative terms are followed by two positive ones. This series is also alternating. Let us find out whether the first condition of the Leibniz test is satisfied.

The requirement is met for all n > 1 . The first Leibniz test is satisfied. Find out if the limit of the common term is equal to zero (using L'Hopital's rule):

We got zero. Thus, both conditions of the Leibniz test are satisfied. Convergence is in place.

Example 5 Investigate the convergence of a series

Solution. This is an alternating row. Let us find out whether the first condition of the Leibniz test is satisfied. Because

Because n ≥ 0 , then 3 n+ 2 > 0 . In turn, for each n, that's why . Consequently, the terms of the series decrease in absolute value. The first Leibniz test is satisfied. Let's find out if the limit of the common term of the series is equal to zero (using L'Hopital's rule):

Received a null value. Both conditions of the Leibniz test are satisfied, so this series converges.

Example 6 Investigate the convergence of a series

Solution. Let us find out whether the first condition of the Leibniz test is satisfied for this alternating series:

The terms of the series decrease in absolute value. The first Leibniz test is satisfied. Find out if the limit of the common term is equal to zero:

The limit of the common term is not equal to zero. The second condition of the Leibniz sign is not fulfilled. Therefore, this series diverges.

The Leibniz sign is a sign conditional convergence of the series. This means that the conclusions about the convergence and divergence of the alternating series considered above can be supplemented: these series converge (or diverge) conditionally.

Absolute convergence of alternating series

Let the row

- alternating. Consider a series composed of the absolute values of its members:

Definition. A series is called absolutely convergent if a series composed of the absolute values of its terms converges. If an alternating series converges, and a series composed of the absolute values of its members diverges, then such an alternating series is called conditionally or not absolutely convergent .

Theorem. If a series converges absolutely, then it converges conditionally.

Example 7 Determine if a series converges

Solution. Corresponding to this series next to the positive terms is the series This generalized harmonic series, where , so the series diverges. Let us check whether the conditions of the Leibniz test are met.

Let's write the absolute values of the first five terms of the series:

As you can see, the terms of the series decrease in absolute value. The first Leibniz test is satisfied. Find out if the limit of the common term is equal to zero:

Received a null value. Both conditions of the Leibniz test are satisfied. That is, on the basis of Leibniz, convergence takes place. And the corresponding series with positive terms diverges. Therefore, this series converges conditionally.

Example 8 Determine if a series converges

absolutely, conditionally, or divergent.

Solution. Corresponding to this series, next to the positive terms, is the series This is a generalized harmonic series, in which, therefore, the series diverges. Let us check whether the conditions of the Leibniz test are satisfied.

Definition

The row is calledalternating if it contains both positive and negative terms.

Example 16 ranks

,
,

are sign-variable.

Alternating series are obviously a special case of alternating series.

For an alternating series the question arises about the connection between its convergence and the convergence of the positive-sign series .

THEOREM 9 (Test for absolute convergence)

If the series converges , then the series converges .

Proof. From the convergence of the series property 3 of convergent series implies the convergence of the series
. Indeed, since
, where
, then the series converges by the first criterion of comparison
.

It follows from this that the series
also converges, since it is the algebraic sum of two convergent series.

In the proved theorem, sufficient sign of convergence row . The converse statement is not true in general.

Definitions

If the series converges , then the row calledabsolutely convergent.

If the row converges, and the series diverges, then the series calledconditionally convergent .

Example 17.
.

The common term of this series
. Because
, then the row
diverges, because it is a Dirichlet series, in which
. Row
converges according to the Leibniz test. Therefore, the series under study converges conditionally.

Example 18. Investigate for convergence series
.

This series converges absolutely, since the series
is a convergent Dirichlet series.

In the study of alternating series for convergence, one can argue according to the following scheme:

Earlier it was noted that in series with positive sign, one can rearrange and group terms arbitrarily. In alternating series, if they converge absolutely, this property is preserved. For conditionally convergent series, the situation is different. Here grouping, permutation of the terms of the series can break the convergence of the series. For example, if positive terms are selected from an alternating conditionally convergent series, then the resulting series may diverge. This circumstance should be borne in mind and handled with great care with conditionally convergent series. For conditionally convergent series, the following Riemann theorem is valid.

THEOREM 10

By changing the order of terms in a conditionally convergent series, it is possible to make its sum equal to any predetermined number and even make the series divergent.

For example, if in a series
permutation of terms, then the series can be represented in the form

So, the sum of the series under consideration is halved. This is because, with conditional convergence, mutual cancellation of positive and negative terms is carried out and, therefore, the sum of the series depends on the order of the terms, while this did not happen with absolute convergence of the series.

Example 19. Investigate for convergence series
.

This series is sign-alternating. Let us examine a series composed of modules of its members, i.e. row
. Using the Cauchy test, we get

Therefore, this series converges absolutely.

Function series Function series and its area of convergence

Let be
,
,...,
,... is a sequence of functions defined on some set
.

Definition

Kind row

, (14)

whose members are functions is calledfunctional .

Giving in (14) various numerical values from the set
, we will get different numerical series. In particular, when
from (14) we get a number series
. This number series can be convergent or divergent. If it converges, then called functional series convergence point (14) .

The set of all points of convergence of a functional series is called itconvergence region and denoted by
. Obviously,
. In particular cases, many
may or may not match the set
or it can also be an empty set. In the latter case, the functional series diverges at each point of the set
.

Area view
for an arbitrary functional series it can be different: the entire numerical axis, interval, union of intervals and half-intervals, etc. In the simplest cases, in the study of functional series for convergence, one can apply the above-considered criteria for the convergence of numerical series, if by x we mean a fixed number.

Definitions

The sum of the first members of the functional series

called
oh partial sum , and the function
, defined in the area
,– the sum of the functional series .

Function defined in scope
, calledrest of the series .

The functional series is calledabsolutely convergent on the set
,if at every point
series converges
.

A series is called alternating if any two of its neighboring members have different signs, i.e. series of the form u 1 – u 2 + u 3 – u 4 +… + u n + …, where u 1 , u 2 , …, u n , … are positive.

Leibniz's theorem. If the terms of the alternating series, taken in absolute value, decrease monotonically and the modulus of the common term of the series tends to zero at , i.e.
, then the series converges.

Example 1

Investigate the convergence of the alternating series:

The terms of the series, taken in absolute value, decrease monotonically:

The series converges.

1.6. Variable rows. Absolute and conditional convergence of a series

Row u 1 + u 2 +…+ u n +… is called alternating if among its members there are both positive and negative.

Alternating series are a special case of alternating series.

Theorem. Given an alternating series u 1 + u 2 +…+ u n +…(1). Let's make a series | u 1 |+| u 2 |+…+| u n |+… (2). If series (2), composed of the absolute values of the terms of series (1), converges, then series (1) converges.

Definition. alternating series u 1 + u 2 +…+ u n +… is called absolutely convergent if the series composed of the absolute values of its members | u 1 |+| u 2 |+…+| u n |+… .

If the alternating series (1) converges, and the series (2), composed of the absolute values of its members, diverges, then this alternating series (1) is called a conditionally or non-absolutely convergent series.

Example 1

Examine the series for convergence and absolute convergence:
.

The alternating series converges according to the Leibniz theorem, because
. The terms of the series decrease monotonically and
. Now we examine this series for absolute convergence. Consider a series composed of the absolute values of the terms of this series: . We investigate the convergence of this series using the d'Alembert test:
. The series converges. Hence, the given alternating series converges absolutely.

Example 2

Examine the series for convergence and absolute convergence:
.

According to Leibniz's theorem
. The series converges. A series composed of the absolute values of the terms of this series has the form
. By the d'Alembert test, we get
. The series converges, which means that the given alternating series converges absolutely.

2. Functional rows. Area of convergence of a functional series

Consider a sequence of functions defined on some interval [ a, b] :

f 1 (x), f 2 (x), f 3 (x) … f n (x), ….

Taking these functions as members of the series, we form the series:

f 1 (x) + f 2 (x) + f 3 (x) + … + f n (x) + …, (1)

which is called functional range.

For example: sin(x) + sin(2x) + sin(3x) + … + sin(nx) + …

In a particular case, the functional series is the series:

which is called power series, where
constant numbers called coefficients of the terms of the power series.

A power series can also be written in the following form:

where
some constant number.

At a certain fixed or numerical value x we get a number series that can be convergent or divergent.

Definition : The set of all values X(or all points X number line) under which the power series converges is called the region of convergence of the power series.

Example 1

Find the region of convergence of the power series:

Solution (1 way).

Let's apply the d'Alembert test.

Since the d'Alembert test is applicable to series only with positive members, then the expression under the limit sign is taken in absolute value.

According to the d'Alembert test, the series converges if
And
.

Those. the series converges if < 1, откуда
or -3< x<3.

Let's get the interval of convergence of this power series: (-3;3).

At the extreme points of the interval x =
, will have
.

In this case, d'Alembert's theorem does not answer the question about the convergence of the series.

We examine the series for convergence at the boundary points:

x=-3,

We get an alternating series. We examine it for convergence on the basis of Leibniz:

1.
the terms of the series, taken in absolute value, decrease monotonically.

2.
Therefore, the series converges at the point x = -3.

x = 3,

We get a positive number. Let us apply the integral Cauchy criterion for the convergence of the series.

the terms of the series decrease monotonically.

Function
in between
:

The improper integral diverges, which means that the series at the point x=3 diverges.

Answer:

Second way determining the area of convergence of a power series is based on the use of the formula for the radius of convergence of a power series:

, where And
odds And
row members.

For this series we have:

. R=3.

series converges

Series convergence interval: -3< x<3.

Further, as in the previous case, it is necessary to investigate at the boundary points: x =
.

Answer: range of convergence of the series [-3;3).

Note that the second way to determine the region of convergence of a power series using the formula for the radius of convergence of the series
more rational.

Example 2

Find the region of convergence of the power series:
.

Let's find R is the convergence radius of the series.

,
,
.

.
.

Series convergence interval (- ;).

We examine the series for convergence at points x = -And x = .

x = - ,

We get an alternating series. We apply the Leibniz sign:

1.
the terms of the series, taken in absolute value, decrease monotonically.

2.
, therefore, the series at the point x = - converges.

x= ,
.

We got a series with positive members. We apply the Cauchy integral test.

Here
:

, members of the series
decrease monotonically.

Function
in between
:

The improper integral diverges, the series diverges.

Answer: [-;) is the region of convergence of the series.

Number series

is called alternating if among its members there are both positive and negative numbers.

A number series is called sign-alternating if any two adjacent members have opposite signs.

where for all (i.e., a series whose positive and negative terms follow each other in turn). For example,

For alternating series, there is a sufficient criterion for convergence (established in 1714 by Leibniz in a letter to I. Bernoulli).

Leibniz sign. Absolute and conditional convergence of a series

Theorem (Leibniz test).

An alternating series converges if:

The sequence of absolute values of the terms of the series monotonically decreases, i.e. ;

The common term of the series tends to zero:.

Moreover, the sum S of the series satisfies the inequalities

Remarks.

Study of an alternating series of the form

(with a negative first term) is reduced by multiplying all its terms by to the study of a series.

Series for which the conditions of Leibniz's theorem are satisfied are called Leibniz series (or Leibniz series).

The relation allows us to obtain a simple and convenient estimate of the error we make by replacing the sum S of a given series by its partial sum.

The discarded series (remainder) is also an alternating series, the sum of which is less than the first term of this series, i.e. Therefore, the error is less than the modulus of the first of the discarded terms.

Example. Calculate the approximate sum of the series.

Solution: given series of Leibniz type. He converges. You can write:

Taking five terms, i.e. replaceable

Let's make a smaller mistake

how. So,.

For alternating series, the following general sufficient criterion for convergence takes place.

Theorem. Let an alternating series be given

If the series converges

composed of the moduli of the members of the given series, then the alternating series itself converges.

The Leibniz convergence criterion for alternating series is a sufficient criterion for the convergence of alternating series.

An alternating series is called absolutely convergent if a series composed of the absolute values of its members converges, i.e. every absolutely convergent series is convergent.

If an alternating series converges, and a series composed of the absolute values of its members diverges, then this series is called conditionally (non-absolutely) convergent.