HIGHER MATHEMATICS

Number series

Lecture.Number series

1. Definition of a number series. Convergence

2. Basic properties of number series

3. Series with positive terms. Signs of convergence

4. Alternating rows. Leibniz convergence test

5. Alternating series

Self-test questions

Literature

Lecture. NUMERIC SERIES

1. Definition of a number series. Convergence.

2. Basic properties of number series.

3. Series with positive terms. Signs of convergence.

4. Alternating rows. Leibniz convergence test.

5. Alternating series.

1. Definition of a number series. Convergence

In mathematical applications, as well as in solving some problems in economics, statistics and other fields, sums with an infinite number of terms are considered. Here we will give a definition of what is meant by such amounts.

Let an infinite number sequence be given

, , …, , …

Definition 1.1. Number series or simply near is called an expression (sum) of the form

. (1.1) are called members of a number, – general or n – m member of the series.

To define series (1.1), it is enough to specify the function of the natural argument

calculating the th term of a series by its number

Example 1.1. Let

. Row (1.2)

called harmonic series .

Example 1.2. Let

, Row (1.3)

called generalized harmonic series. In the special case when

a harmonic series is obtained.

Example 1.3. Let

= . Row (1.4)

called near geometric progression.

From the terms of series (1.1) we form a numerical sequence of partialsamounts Where

– the sum of the first terms of the series, which is called n-th partial amount, i.e. , , ,

…………………………….

, (1.5)

…………………………….

Number sequence

with an unlimited increase in number it can:

1) have a finite limit;

2) have no finite limit (the limit does not exist or is equal to infinity).

Definition 1.2. Series (1.1) is called convergent, if the sequence of its partial sums (1.5) has a finite limit, i.e.

In this case the number

called amount series (1.1) and is written .

Definition 1.3.Series (1.1) is called divergent, if the sequence of its partial sums does not have a finite limit.

No sum is assigned to the divergent series.

Thus, the problem of finding the sum of a convergent series (1.1) is equivalent to calculating the limit of the sequence of its partial sums.

Let's look at a few examples.

Example 1.4. Prove that the series

converges and find its sum.

We'll find n- th partial sum of this series

.

General member

Let's represent the series in the form .

From here we have:

. Therefore, this series converges and its sum is equal to 1:

Example 1.5. Examine the series for convergence

(1.6)

For this row

. Therefore, this series diverges.

Comment. At

series (1.6) is the sum of an infinite number of zeros and is obviously convergent.

Example 1.6. Examine the series for convergence

(1.7)

For this row

In this case, the limit of a sequence of partial sums is

does not exist, and the series diverges.

Example 1.7. Examine the series of geometric progression (1.4) for convergence:

It is easy to show that n-th partial sum of a geometric progression series at

is given by the formula.

Let's consider the cases:

Then and.

Therefore, the series converges and its sum is equal to

1 property.

Dropping a finite number of terms does not affect the convergence of the equation.

ConsiderLet

If there is a finite limit on the right in (29.1), then there is also a limit on the left, and the series converges

2 property.

If the series converges and has sum S, then the series

c = const, converges and has the sum cS.

Let then

3 property.

If the series converge and have sums, then the series converge and have sum

1. Series with positive terms. Signs for comparing the convergence of positive series. Positive series

If a n ≥ 0 (n= 1, 2, 3, ...), then the series a 1 +a 2 +a 3 + ... is called positive. In the case when in front of everyone n turns out a n> 0, we will call the series strictly positive.

Positive series have many properties that make them similar to ordinary sums of a finite number of terms.

It is easy to see that the partial sum S n =a 1 +a 2 + ... +a n positive series increases(maybe not strictly) with increasing n. Since every increasing number sequence has a finite or infinite limit (and the terms of the sequence do not exceed this limit), then for any positive series there is a limit

This limit will be finite or infinite, depending on whether the set of partial sums is bounded above or not ( S n). Thus, there is

Theorem 1. A positive series converges if and only if the set of its partial sums is bounded above.

Of course, for a non-positive series, the boundedness of the set of partial sums does not ensure convergence, as can be seen from the example of the series 1 + (-1) + 1 + (-1) + ...

We also note that the partial sums of a convergent positive series do not exceed its sum.

The theorem proved reduces the question of the convergence of a positive series to the simpler question of the boundedness of the set of its partial sums.

Consider, for example, the series (24)

in which α > 1. The sum of this series can be written as follows:

Since the sum contains 2 k terms, and the largest of them is the first, then this sum does not exceed the number

That's why

The sum on the right here is the partial sum of the geometric progression

As was proven earlier, this progression converges (since α > 1), and its sum is equal to

Since progression (25) is also a positive series, its partial sums do not exceed its sum (26). Especially

This inequality is established for any m. But for everyone n you can find something like this m that 2 m - 1 >n.

Therefore, in any case n It turns out that series (24) converges.

It should be noted, however, that direct application of Theorem 1 is relatively rare.

Usually, based on it, but more convenient tests for the convergence of series are used. The simplest of them is the so-called series comparison sign

If each member of a positive series is not greater than the member of another series having the same number, then the second series is called majorant in relation to the first.

In other words, a series b 1 +b 2 +b 3 + ... is majorant with respect to the series a 1 +a 2 +a 3 + ..., if for all n will a n ≤b n .

It is easy to understand that the partial sum of a given series is not greater than (having the same number) the partial sum of a majorant series. This means that if the partial sums of a majorant series are bounded from above, then this is even more so for the original series. It follows from this

Theorem 2. If for a positive series there is a convergent majorant series, then this series itself converges. If a given series diverges, then every majorant series for it diverges.

Consider, for example, the series (27)

assuming α < 1. Ясно, что этот ряд - мажорантный по отношению к гармоническому ряду, и потому ряд (27) расходится.

The first sign of comparison of series. Let and be two positive number series and the inequality holds for all k = 1, 2, 3, ... Then the convergence of the series implies convergence, and the divergence of the series implies divergence. The first comparison criterion is used very often and is a very powerful tool for studying number series for convergence. The main problem is selecting a suitable series for comparison. The series for comparison is usually (but not always) chosen so that its exponent kth term is equal to the difference between the exponents of the numerator and denominator kth member of the number series under study. For example, let the difference between the exponents of the numerator and denominator be equal to 2 – 3 = -1 , therefore, for comparison we select the row with kth member, that is, a harmonic series. Let's look at a few examples. Example. Determine the convergence or divergence of a series. Solution. Since the limit of the general term of the series is zero, the necessary condition for the convergence of the series is satisfied. It is easy to see that the inequality is true for all natural k. We know that the harmonic series diverges; therefore, by the first criterion of comparison, the original series is also divergent. Example. Explore numerical series convergence. Solution. The necessary condition for the convergence of a number series is satisfied, since . The inequality is obvious for any natural value k. The series converges, since the generalized harmonic series is convergent for s > 1. Thus, the first sign of comparison of series allows us to state the convergence of the original number series. Example. Determine the convergence or divergence of a number series. Solution., therefore, the necessary condition for the convergence of the number series is satisfied. Which row should I choose for comparison? A number series suggests itself, but in order to decide on s, carefully examine the number sequence. The terms of the number sequence increase towards infinity. Thus, starting from some number N(namely, with N=1619), the members of this sequence will be greater 2 . Starting from this number N, the inequality is true. A number series converges due to the first property of convergent series, since it is obtained from a convergent series by discarding the first N – 1 member. Thus, by the first property of comparison, the series is convergent, and by virtue of the first property of convergent number series, the series will also converge. The second sign of comparison. Let them be positive-sign number series. If, then convergence of the series implies convergence. If, then divergence follows from the divergence of a number series. Consequence. If and, then from the convergence of one series the convergence of the other follows, and from the divergence the divergence follows. We study series convergence using the second comparison criterion. Let's take a convergent series as a series. Let's find the limit of the ratio k's members of number series: Thus, according to the second criterion of comparison, from the convergence of a numerical series, the convergence of the original series follows.

Example. Examine the convergence of a number series. Solution. Let us check the necessary condition for the convergence of the series . The condition is met. To apply the second comparison criterion, let's take a harmonic series. Let's find the limit of the ratio k's members: Consequently, from the divergence of the harmonic series, the divergence of the original series according to the second criterion of comparison follows. For information, we present the third criterion for comparing series. The third sign of comparison. Let them be positive-sign number series. If from a certain number N the condition is satisfied, then convergence of the series implies convergence, and divergence of the series implies divergence.

1. Number series: basic concepts, necessary conditions for the convergence of the series. The rest of the row.

2. Series with positive terms and tests of their convergence: tests of comparison, D'Alembert, Cauchy.

3. Alternating series, Leibniz’s test.

1. Definition of a number series. Convergence

In mathematical applications, as well as in solving some problems in economics, statistics and other fields, sums with an infinite number of terms are considered. Here we will give a definition of what is meant by such amounts.

Let an infinite number sequence be given

Definition 1.1. Number series or simply near is called an expression (sum) of the form

. (1.1)

Numbers are called members of a number, –general or n–m member of the series.

To define series (1.1), it is enough to specify the function of the natural argument of calculating the th term of the series by its number

Example 1.1. Let . Row

(1.2)

called harmonic series.

Example 1.2. Let ,Row

(1.3)

called generalized harmonic series. In a particular case, a harmonic series is obtained.

Example 1.3. Let =. Row

called near geometric progression.

From the terms of series (1.1) we form a numerical sequence of partials amounts Where – the sum of the first terms of the series, which is called n-th partial amount, i.e.

…………………………….

…………………………….

Number sequence with an unlimited increase in number, it can:

1) have a finite limit;

2) have no finite limit (the limit does not exist or is equal to infinity).

Definition 1.2. Series (1.1) is called convergent, if the sequence of its partial sums (1.5) has a finite limit, i.e.

In this case the number is called amount series (1.1) and is written

Definition 1.3. Series (1.1) is called divergent, if the sequence of its partial sums does not have a finite limit.

No sum is assigned to the divergent series.

Thus, the problem of finding the sum of a convergent series (1.1) is equivalent to calculating the limit of the sequence of its partial sums.

Let's look at a few examples.

Example 1.4. Prove that the series

converges and find its sum.

Let's find the nth partial sum of this series.

General member represent the series in the form .

From here we have: . Therefore, this series converges and its sum is equal to 1:

Example 1.5. Examine the series for convergence

For this row

. Therefore, this series diverges.

Comment. For series (1.6) is the sum of an infinite number of zeros and is obviously convergent.

2. Basic properties of number series

The properties of a sum of a finite number of terms differ from the properties of a series, i.e., the sum of an infinite number of terms. So, in the case of a finite number of terms, they can be grouped in any order, this will not change the sum. There are convergent series (conditionally convergent, which will be considered in Section 5), for which, as Riemann showed * , by appropriately changing the order of their terms, you can make the sum of the series equal to any number, and even a divergent series.

Example 2.1. Consider a divergent series of the form (1.7)

By grouping its members in pairs, we obtain a convergent number series with a sum equal to zero:

On the other hand, by grouping its terms in pairs, starting with the second term, we also obtain a convergent series, but with a sum equal to one:

Convergent series have certain properties that make it possible to treat them as if they were finite sums. So they can be multiplied by numbers, added and subtracted term by term. They can combine any adjacent terms into groups.

Theorem 2.1.(A necessary sign of convergence of a series).

If series (1.1) converges, then its common term tends to zero as n increases indefinitely, i.e.

The proof of the theorem follows from the fact that , and if

S is the sum of series (1.1), then

Condition (2.1) is a necessary but not sufficient condition for the convergence of the series. That is, if the common term of the series tends to zero at , this does not mean that the series converges. For example, for the harmonic series (1.2) however, as will be shown below, it diverges.

Consequence(A sufficient sign of the divergence of the series).

If the common term of a series does not tend to zero at, then this series diverges.

Example 2.2. Examine the series for convergence

.

For this row

Therefore, this series diverges.

The divergent series (1.6), (1.7) considered above are also such due to the fact that the necessary convergence criterion is not satisfied for them. For series (1.6), the limit for series (1.7) the limit does not exist.

Property 2.1. The convergence or divergence of a series will not change if a finite number of terms are arbitrarily removed from it, added to it, or rearranged in it (in this case, for a convergent series, its sum may change).

The proof of the property follows from the fact that series (1.1) and any of its remainders converge or diverge simultaneously.

Property 2.2. A convergent series can be multiplied by a number, i.e., if the series (1.1) converges, has the sum S and c is a certain number, then

The proof follows from the fact that the following equalities hold for finite sums:

Property 2.3. Convergent series can be added and subtracted term by term, i.e. if the series,

converge,

converges and its sum is equal to i.e.

.

The proof follows from the properties of the limit of finite sums, i.e.

1. If a 1 + a 2 + a 3 +…+a n +…= converges, then the series a m+1 +a m+2 +a m+3 +…, obtained from this series by discarding the first m terms, also converges. This resulting series is called the mth remainder of the series. And, vice versa: from the convergence of the mth remainder of the series, the convergence of this series follows. Those. The convergence and divergence of a series is not violated if a finite number of its terms are added or discarded.

2 . If the series a 1 + a 2 + a 3 +... converges and its sum is equal to S, then the series Ca 1 + Ca 2 +..., where C = also converges and its sum is equal to CS.

3. If the series a 1 +a 2 +... and b 1 +b 2 +... converge and their sums are equal to S1 and S2, respectively, then the series (a 1 +b 1)+(a 2 +b 2)+(a 3 +b 3)+… and (a 1 -b 1)+(a 2 -b 2)+(a 3 -b 3)+… also converge. Their sums are respectively equal to S1+S2 and S1-S2.

4. A). If a series converges, then its nth term tends to 0 as n increases indefinitely (the converse is not true).

- necessary sign (condition)convergence row.

b). If
then the series is divergent - sufficient conditiondivergences row.

-series of this type are studied only according to property 4. This divergent rows.

Sign-positive series.

Signs of convergence and divergence of positive-sign series.

Positive series are series in which all terms are positive. We will consider these signs of convergence and divergence for series with positive signs.

1. The first sign of comparison.

Let two positive-sign series a 1 + a 2 + a 3 +…+a n +…= be given (1) иb 1 +b 2 +b 3 +…+b n +…= (2).

If the members of the series (1) not moreb n and series (2) converges, then series (1) also converges.

If the members of the series (1) not less corresponding members of series (2), i.e. a n b n and row (2) diverges, then series (1) also diverges.

This comparison criterion is valid if the inequality is not satisfied for all n, but only starting from some.

2. Second sign of comparison.

If there is a finite and non-zero limit
, then both series converge or diverge simultaneously.

- rows of this type diverge according to the second criterion of comparison. They must be compared with the harmonic series.

3. D'Alembert's sign.

If for a positive series (a 1 + a 2 + a 3 +…+a n +…= ) exists
(1), then the series converges if q<1, расходится, если q>

4. Cauchy's sign is radical.

If there is a limit for a positive series
(2), then the series converges ifq<1, расходится, если q>1. If q=1 then the question remains open.

5. Cauchy's test is integral.

Let us recall improper integrals.

If there is a limit
. This is an improper integral and is denoted
.

If this limit is finite, then the improper integral is said to converge. The series, respectively, converges or diverges.

Let the series a 1 + a 2 + a 3 +…+a n +…= - positive series.

Let us denote a n =f(x) and consider the function f(x). If f(x) is a positive, monotonically decreasing and continuous function, then if the improper integral converges, then the given series converges. And vice versa: if the improper integral diverges, then the series diverges.

If the series is finite, then it converges.

Rows are very common
-Derichlet series. It converges if p>1, diverges p<1. Гармонический ряд является рядом Дерихле при р=1. Сходимость и расходимость данного ряда легко доказать с помощью интегрального признака Коши.

1. Basic concepts. Let us be given an infinite sequence of numbers

Definition. Expression

where is the common term of the series.

Example 7.1

Let's consider the series. Here is the common term of the series.

Let us consider the sums made up of a finite number of terms of the series (7.1): , , , ..., , . . . Such amounts are called partial amounts row. is called the th partial sum of the series. Thus, a partial sum is the sum of (a finite number of) terms:

.

(7.3)

Subsequence , , , ..., , ... or .is called a sequence of partial sums of the series (7.1).

Definition. If there is a finite limit , then series (1.1) is called convergent, and the number is the sum of this series. In this case they write

If the sequence has no limit, then the series (7.1) is called divergent. A divergent series has no sum.

Example 7.2

Solution

The general term of the series can be represented as

, (n= 1, 2, 3, . . .).

Therefore, this series converges and its sum is 1.

Example 7.3(geometric progression)

Consider a sequence, each term of which, starting from the second, is obtained by multiplying the previous term by the same number:

Sometimes the series (7.5) itself is called a geometric progression.

The partial sum of series (7.5) is the sum of the terms of the geometric progression and

calculated by the formula

.

(7.6)

If, then. Consequently, when the series (7.5) converges. If, then. Consequently, when series (7.5) diverges. If , then (7.5) turns into the series 1 + 1 + 1 + ... + 1 + ... . For such a series and

Consequently, when series (7.5) diverges.

When considering series, the issue of convergence (divergence) is important. To address this issue, Examples 7.1 and 7.2 used the definition of convergence. More often, certain properties of the series are used for this, which are called signs of convergence of the series.

Theorem 7.1(a necessary sign of convergence). If the series (7.1) converges, then its common term tends to zero with an unlimited increase in , i.e.

The series (7.8) is called harmonic near.

For this row. However, no conclusion about the convergence of series (7.8) can yet be made, since the statement converse to Theorem 7.1 is not true.

Let us show that series (7.8) diverges. This can be established by contradictory reasoning. Suppose that series (7.8) converges and its sum is equal to S.Then = –

– , which contradicts the inequality

Consequently, the harmonic series diverges.

The necessary feature can be used to establish the fact of divergence of a series. Indeed, it follows from Theorem 7.1 that if the common term of the series does not tend to zero, then the series diverges.

Example 7.5

Let's consider the series.

Here , . The limit is not equal to zero, therefore the series diverges.

Thus, if condition (7.7) is satisfied, the question of the convergence of series (7.1) remains open. The series may diverge, or it may converge. To resolve this issue they can

the properties of the series must be used, from which the convergence of this series follows. Such properties are called sufficient signs of convergence rows.

Series with positive terms. Consider sufficient signs of convergence of series with positive terms.

Theorem 7.2.(D'Alembert's sign).

are positive:

1) if , series (7.1) converges;

2) if , series (7.1) converges;

Note. Series (7.1) will also diverge in the case when , since then, starting from some number N, will be and, therefore, does not tend to zero at .

Example 7.6

Examine the series for convergence.

Solution. . . then =

The found limit is less than unity. Therefore, this series converges.

Example 7.7

Examine the series for convergence.

Solution. . . then =

= = = = = = = .

The found limit is greater than unity. Therefore, this series diverges.

Theorem 7.3.(Radical Cauchy sign).

Let a series (7.1) be given, all terms of which are positive:

and there is a limit

,

(7.11)

(where is the designation of the found limit). Then:

1) if , series (7.1) converges;

2) if , series (7.1) converges;

3) if , the criterion under consideration does not answer the question about the convergence of the series.

A proof of the sign can be found in.

Example 7.8

Examine the series for convergence.

Solution.

Let's find the limit (7.11):

The found limit is greater than unity. Consequently, this series diverges (Theorem 7.3).

Generalized harmonic series.Generalized harmonic series called a series of the form

Theorem 7.3. (Leibniz's theorem). If for a series(7.13) two conditions are met:

1) the terms of the series decrease monotonically in absolute value:

2)the common term of the series tends to zero:

then a series(7.13) converges.

A proof of the sign can be found, for example, in.

Example 7.9.

Consider the sign of the alternating series

(7.14)

For this series, the conditions of Theorem (7.13) are satisfied:

Consequently, series (7.12) converges.

Corollary to Theorem 7.3. The remainder of the alternating series (7.13), which satisfies the conditions of Leibniz’s theorem, has the sign of its first term and is less than it in absolute value.

Example 7.10. Calculate the sum of a convergent series with an accuracy of 0.1

As an approximate value of the sum of the series, we must take the partial sum for which . According to the investigation, . Therefore, it is enough to put , i.e., then

Hence, with an accuracy of 0.1.

Absolute and conditional convergence. Consider a series whose terms have arbitrary signs

Note that the series (7.16) is a series with positive terms and the corresponding theorems given above are applicable to it.

Theorem 7.4(A sign of absolute convergence). If the series (7.16) converges, then the series (7.15) also converges.

(The proof of the theorem can be found, for example, in).

Definition.

If the series (7.16) converges, then the corresponding series (7.15) is called absolutely convergent absolutely descending Xia.

It may turn out that series (7.16) diverges, but series (7.15) converges. In this case, series (7.15) is called conditionally convergent.

Note that the alternating series (7.13) is a special case of a series whose terms have arbitrary signs. Therefore, to study an alternating series, we can also apply Theorem 7.5.

Example 7.11

Solution

Let's consider a series made up of the absolute values ​​of the members of a given series. This series converges, because it is a generalized harmonic series (7.12) with the value Therefore, according to the absolute convergence criterion (Theorem 7.5), the original series converges absolutely.

Example 7.12

The series is examined for convergence.

Solution

according to Leibniz's theorem, it converges, but the series composed of the absolute values ​​of the terms of the original series diverges (this is a harmonic series). Consequently, the original series converges conditionally.