Rows for dummies. Examples of solutions

I welcome all survivors to the second year! In this lesson, or rather, in a series of lessons, we will learn how to manage rows. The topic is not very complicated, but mastering it will require knowledge from the first year, in particular, you need to understand what is a limit, and be able to find the simplest limits. However, it’s okay, as I explain, I will provide relevant links to the necessary lessons. To some readers, the topic of mathematical series, solution methods, signs, theorems may seem peculiar, and even pretentious, absurd. In this case, you don’t need to be too “loaded”; we accept the facts as they are and simply learn to solve typical, common tasks.

1) Rows for dummies, and for samovars immediately content :)

For super-fast preparation on the topic There is an express course in pdf format, with the help of which you can really “raise” your practice literally in a day.

The concept of a number series

IN general view number series can be written like this: .
Here:
– mathematical sum icon;
– common term of the series(remember this simple term);
– “counter” variable. The notation means that summation is carried out from 1 to “plus infinity”, that is, first we have , then , then , and so on - to infinity. Instead of a variable, a variable or is sometimes used. Summation does not necessarily start from one; in some cases it can start from zero, from two, or from any natural number.

In accordance with the “counter” variable, any series can be expanded:
- and so on, ad infinitum.

Components - This NUMBERS which are called members row. If they are all non-negative (greater than or equal to zero), then such a series is called positive number series.

Example 1

This, by the way, is already a “combat” task - in practice, quite often it is necessary to write down several terms of a series.

First, then:
Then, then:
Then, then:

The process can be continued indefinitely, but according to the condition it was required to write the first three terms of the series, so we write down the answer:

Please note the fundamental difference from number sequence,
in which the terms are not summed up, but are considered as such.

Example 2

Write down the first three terms of the series

This is an example for independent decision, answer at the end of the lesson

Even for a series that is complex at first glance, it is not difficult to describe it in expanded form:

Example 3

Write down the first three terms of the series

In fact, the task is performed orally: mentally substitute into the common term of the series first, then and. Eventually:

We leave the answer as follows: It is better not to simplify the resulting series terms, that is do not perform actions: , , . Why? The answer is in the form it is much easier and more convenient for the teacher to check.

Sometimes the opposite task occurs

Example 4

There is no clear solution algorithm here, you just need to see the pattern.
In this case:

To check, the resulting series can be “written back” in expanded form.

Here's an example that's a little more complicated to solve on your own:

Example 5

Write down the sum in collapsed form with the common term of the series

Perform a check by again writing the series in expanded form

Convergence of number series

One of the key objectives of the topic is study of series for convergence. In this case, two cases are possible:

1) Rowdiverges. This means that an infinite sum is equal to infinity: or sums in general does not exist, as, for example, in the series
(here, by the way, is an example of a series with negative terms). A good example of a divergent number series was found at the beginning of the lesson: . Here it is quite obvious that each next member of the series is greater than the previous one, therefore and, therefore, the series diverges. An even more trivial example: .

2) Rowconverges. This means that an infinite sum is equal to some finite number: . Please: – this series converges and its sum is zero. As a more meaningful example, we can cite infinitely decreasing geometric progression, known to us since school: . The sum of the terms of an infinitely decreasing geometric progression is calculated using the formula: , where is the first term of the progression, and is its base, which is usually written in the form correct fractions In this case: , . Thus: A finite number is obtained, which means the series converges, which is what needed to be proved.

However, in the vast majority of cases find the sum of the series is not so simple, and therefore in practice, to study the convergence of a series, special signs that have been proven theoretically are used.

There are several signs of series convergence: necessary test for the convergence of a series, comparison tests, D'Alembert's test, Cauchy's tests, Leibniz's sign and some other signs. When to use which sign? It depends on the common member of the series, figuratively speaking, on the “filling” of the series. And very soon we will sort everything out.

! To further learn the lesson, you must understand well what is a limit and it is good to be able to reveal the uncertainty of a type. To review or study the material, please refer to the article Limits. Examples of solutions.

A necessary sign of convergence of a series

If a series converges, then its common term tends to zero: .

Reverse to general case false, i.e., if , then the series can either converge or diverge. And therefore this sign is used to justify divergences row:

If the common term of the series does not tend to zero, then the series diverges

Or in short: if , then the series diverges. In particular, a situation is possible where the limit does not exist at all, as, for example, limit. So they immediately justified the divergence of one series :)

But much more often, the limit of a divergent series is equal to infinity, and instead of “x” it acts as a “dynamic” variable. Let's refresh our knowledge: limits with “x” are called limits of functions, and limits with the variable “en” are called limits of numerical sequences. The obvious difference is that the variable "en" takes discrete (discontinuous) natural values: 1, 2, 3, etc. But this fact has little effect on methods for solving limits and methods for disclosing uncertainties.

Let us prove that the series from the first example diverges.
Common member of the series:

Conclusion: row diverges

The necessary feature is often used in real practical tasks:

Example 6

We have polynomials in the numerator and denominator. The one who carefully read and comprehended the method of disclosing uncertainty in the article Limits. Examples of solutions, I probably caught that when the highest powers of the numerator and denominator equal, then the limit is finite number .

Divide the numerator and denominator by

Series under study diverges, since the necessary criterion for the convergence of the series is not fulfilled.

Example 7

Examine the series for convergence

This is an example for you to solve on your own. Full solution and answer at the end of the lesson

So, when we are given ANY number series, Firstly we check (mentally or on a draft): does its common term tend to zero? If it doesn’t, we formulate a solution based on examples No. 6, 7 and give an answer that the series diverges.

What types of apparently divergent series have we considered? It is immediately clear that series like or diverge. The series from examples No. 6, 7 also diverge: when the numerator and denominator contain polynomials, and the leading power of the numerator is greater than or equal to the leading power of the denominator. In all these cases, when solving and preparing examples, we use the necessary sign of convergence of the series.

Why is the sign called necessary? Understand in the most natural way: in order for a series to converge, necessary, so that its common term tends to zero. And everything would be great, but there’s more not enough. In other words, if the common term of a series tends to zero, THIS DOES NOT MEAN that the series converges– it can both converge and diverge!

Meet:

This series is called harmonic series. Please remember! Among the number series, he is a prima ballerina. More precisely, a ballerina =)

It's easy to see that , BUT. In the theory of mathematical analysis it has been proven that harmonic series diverges.

You should also remember the concept of a generalized harmonic series:

1) This row diverges at . For example, the series , , diverge.
2) This row converges at . For example, the series , , , converge. I emphasize once again that in almost all practical tasks it is not at all important to us what the sum of, for example, the series is equal to, the very fact of its convergence is important.

These are elementary facts from the theory of series that have already been proven, and when solving any practical example one can safely refer, for example, to the divergence of a series or the convergence of a series.

In general, the material in question is very similar to study of improper integrals, and it will be easier for those who have studied this topic. Well, for those who haven’t studied it, it’s doubly easier :)

So, what to do if the common term of the series TENDS to zero? In such cases, to solve examples you need to use others, sufficient signs of convergence/divergence:

Comparison criteria for positive number series

I draw your attention, that here we are talking only about positive number series (with non-negative terms).

There are two signs of comparison, one of them I will simply call a sign of comparison, another - limit of comparison.

Let's first consider comparison sign, or rather, the first part of it:

Consider two positive number series and . If known, that the series – converges, and, starting from some number, the inequality is satisfied, then the series also converges.

In other words: From the convergence of the series with larger terms follows the convergence of the series with smaller terms. In practice, the inequality often holds for all values:

Example 8

Examine the series for convergence

First, let's check(mentally or in draft) execution:
, which means it was not possible to “get off with little blood.”

We look into the “pack” of the generalized harmonic series and, focusing on the highest degree, we find a similar series: It is known from theory that it converges.

For all natural numbers, the obvious inequality holds:

and larger denominators correspond to smaller fractions:
, which means, based on the comparison criterion, the series under study converges together with next to .

If you have any doubts, you can always describe the inequality in detail! Let us write down the constructed inequality for several numbers “en”:
If , then
If , then
If , then
If , then
….
and now it is absolutely clear that inequality fulfilled for all natural numbers “en”.

Let's analyze the comparison criterion and the solved example from an informal point of view. Still, why does the series converge? Here's why. If a series converges, then it has some final amount: . And since all members of the series less corresponding terms of the series, then it is clear that the sum of the series cannot be greater than the number, and even more so, cannot be equal to infinity!

Similarly, we can prove the convergence of “similar” series: , , etc.

! note, that in all cases we have “pluses” in the denominators. The presence of at least one minus can seriously complicate the use of the product in question. comparison sign. For example, if a series is compared in the same way with a convergent series (write out several inequalities for the first terms), then the condition will not be satisfied at all! Here you can dodge and select another convergent series for comparison, for example, but this will entail unnecessary reservations and other unnecessary difficulties. Therefore, to prove the convergence of a series it is much easier to use limit of comparison(see next paragraph).

Example 9

Examine the series for convergence

And in this example, I suggest you consider for yourself second part of the comparison attribute:

If known, that the series – diverges, and starting from some number (often from the very first), the inequality is satisfied, then the series also diverges.

In other words: From the divergence of a series with smaller terms follows the divergence of a series with larger terms.

What should be done?
It is necessary to compare the series under study with a divergent harmonic series. For a better understanding, construct several specific inequalities and make sure that the inequality is fair.

The solution and sample design are at the end of the lesson.

As already noted, in practice, the comparison criterion just discussed is rarely used. The real workhorse of number series is limit of comparison, and in terms of frequency of use it can only compete with d'Alembert's sign.

Limit test for comparing numerical positive series

Consider two positive number series and . If the limit of the ratio of the common terms of these series is equal to finite non-zero number: , then both series converge or diverge simultaneously.

When is the limiting criterion used? The limiting criterion for comparison is used when the “filling” of the series is polynomials. Either one polynomial in the denominator, or polynomials in both the numerator and denominator. Optionally, polynomials can be located under the roots.

Let's deal with the row for which the previous comparison sign has stalled.

Example 10

Examine the series for convergence

Let's compare this series with a convergent series. We use the limiting criterion for comparison. It is known that the series converges. If we can show that equals finite, non-zero number, it will be proven that the series also converges.

A finite non-zero number is obtained, which means the series under study is converges together with next to .

Why was the series chosen for comparison? If we had chosen any other series from the “cage” of the generalized harmonic series, then we would not have succeeded in the limit finite, non-zero numbers (you can experiment).

Note: when we use the limiting comparison criterion, doesn't matter, in what order to compose the relation of common members, in the example considered, the relation could be compiled the other way around: - this would not change the essence of the matter.

There is no finite limit to partial sums. For example, rows

diverge.

R.r. began to appear in the works of mathematicians of the 17th and 18th centuries. L. Euler was the first to come to the conclusion that it is necessary to pose the question, not what the amount is equal to, but how to determine the amount of R. R., and found an approach to solving this question that is close to the modern one. R.r. to the end 19th century found no use and were almost forgotten. Accumulation towards the end 19th century various facts of mathematics. analysis reawakened interest in R. r. The question began to be raised about the possibility of summing series in a certain sense different from the usual one.

EXAMPLES. 1) If you multiply two rows

converging respectively to A and B, then the series obtained as a result of multiplication

may turn out to be divergent. However, if the sum of series (1) is determined not as partial sums s n, but as

(2)

then in this sense, series (1) will always converge (i.e., the limit in (2) will exist) and its sum in this sense is equal to C=AB.

2) Fourier series of the function f(x) , continuous at the point x 0 (or having a discontinuity of the 1st kind), can diverge at this point. If the sum of the series is determined by formula (2), then in this sense the Fourier series of such a function will always converge and its sum in this sense is equal to f(x 0) (or, accordingly, if x 0 - discontinuity point of the 1st kind).

3) Power series

converges for to the sum and diverges for . If the sum of the series is defined as

(4)

Where s n are partial sums of series (3), then in this sense series (3) will converge for all z satisfying the condition Re z

To generalize the concept of the sum of a series in the theory of R. R. consider a certain operation or rule, as a result of which R. r. is placed in a certain, called. its sum (in this definition). This rule is called summation method. So, the rule described in example 1), called. by the method of summing arithmetic averages (see Cesaro summation methods). The rule defined in example 2) is called. Borel's summation method.

see also Summation of divergent series. Lit.: Vogue 1 E., Lecons sur les series divergentes, P., 1928; Hard and G., Divergent Series, trans. from English, M., 1951; Cook R., Infinite matrices and sequence spaces, trans. from English, M., I960; R e u e r i m h o f f A., Lectures on summability, V., 1969; K n o r K., Theory and application on infinite series, N. Y., 1971; Z e 1 1 e r K., B e e k m a n n V., Theory der Limitierungsverfahren, B.- Hdlb. - N.Y., 1970. I. I. Volkov.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

See what “DIVERGING SERIES” is in other dictionaries:

divergent series- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Topics of energy in general EN divergent series ... Technical Translator's Guide

divergent series- diverguojančioji eilutė statusas T sritis fizika atitikmenys: engl. divergent series vok. divergente Reihe, f rus. divergent series, m pranc. série divergente, f … Fizikos terminų žodynas

A series in which the sequence of partial sums does not have a finite limit. If the common term of the series does not tend to zero, then the series diverges, for example 1 1 + 1 1 + ... + (1) n 1 + ...; example of R. p., the general term of which tends to zero,... ...

Adding terms of the Fourier series ... Wikipedia

A series, an infinite sum, for example of the form u1 + u2 + u3 +... + un +... or, in short, . (1) One of the simplest examples of a sequence, found already in elementary mathematics, is the sum of an infinitely decreasing geometric progression 1 + q + q 2 +... + q... ... Great Soviet Encyclopedia

Content. 1) Definition. 2) A number determined by a series. 3) Convergence and divergence of series. 4) Conditional and absolute convergence. 5) Uniform convergence. 6) Expansion of functions into series. 1. Definitions. R. is a sequence of elements... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Ephron

I is an infinite sum, for example, of the form u1 + u2 + u3 +... + un +... or, in short, One of the simplest examples of sum, found already in elementary mathematics, is an infinitely decreasing sum... ... Great Soviet Encyclopedia

An infinite sum, a sequence of elements (called members of a given series) of a certain linear topological. space and a certain infinite set of their finite sums (called partial sums of the world... ... Mathematical Encyclopedia

Fourier series representation of an arbitrary function f with period τ in the form of a series. This series can also be rewritten in the form. where Ak is the amplitude of the kth harmonic oscillation (cos function), circle ... Wikipedia

Definition 1.1. A number series with a common term is a sequence of numbers connected by an addition sign, i.e. an expression of the form:

This series can also be written in the form

Example 1.1. If then the series looks like:

Sometimes when writing a series, only its first few members are written down. This is done only when the pattern characteristic of the members of the series is easily discernible. Strictly speaking, this method of specifying a series is not mathematically correct, since obtaining a formula for the general term from the first few terms of a series is a problem that does not have a unique solution.

Example 1.2. Let's write one of the possible formulas for the general term of the series, knowing its first 4 terms:

Solution. Let us first consider the sequence of numerators 2, 5, 8, 11. They form an arithmetic progression, the first term of which is 2, and the difference is 3. This allows us to take the formula for the general term of the arithmetic progression as a general expression for the numerator: Denominators 2, 6, 18, 54 form a geometric progression with

the first term is 2 and the denominator is 3. As their general expression, we can take the formula for the general term of a geometric progression. So, the general term of the series will have the following form:

It should be noted that a more complex expression could be taken as a general term

Basic definitions.

Definition. The sum of the terms of an infinite number sequence is called number series.

At the same time, the numbers
we will call them members of the series, and u n– a common member of the series.

Definition. Amounts
,n = 1, 2, … are called private (partial) amounts row.

Thus, it is possible to consider sequences of partial sums of the series S 1 , S 2 , …, S n , …

Definition. Row
called convergent, if the sequence of its partial sums converges. Sum of convergent series is the limit of the sequence of its partial sums.

Definition. If the sequence of partial sums of a series diverges, i.e. has no limit, or has an infinite limit, then the series is called divergent and no amount is assigned to it.

Properties of rows.

1) The convergence or divergence of the series will not be violated if you change, discard or add a finite number of terms of the series.

2) Consider two rows
And
, where C is a constant number.

Theorem. If the row
converges and its sum is equalS, then the series
also converges, and its sum is equal to CS. (C  0)

3) Consider two rows
And
.Amount or difference of these series will be called a series
, where the elements are obtained by adding (subtracting) the original elements with the same numbers.

Theorem. If the rows
And
converge and their sums are equal respectivelySAnd , then the series
also converges and its sum is equalS +  .

The difference of two convergent series will also be a convergent series.

The sum of a convergent and a divergent series is a divergent series.

It is impossible to make a general statement about the sum of two divergent series.

When studying series, they mainly solve two problems: studying convergence and finding the sum of the series.

Cauchy criterion.

(necessary and sufficient conditions for the convergence of the series)

In order for the sequence
was convergent, it is necessary and sufficient that for any
there was such a numberN, that atn > Nand anyp> 0, where p is an integer, the following inequality would hold:

.

Proof. (necessity)

Let
, then for any number
there is a number N such that the inequality

is fulfilled when n>N. For n>N and any integer p>0 the inequality also holds
. Taking into account both inequalities, we obtain:

The need has been proven. We will not consider the proof of sufficiency.

Let us formulate the Cauchy criterion for the series.

In order for the series
was convergent, it is necessary and sufficient that for any
there was a numberNsuch that atn> Nand anyp>0 the inequality would hold

.

However, in practice, using the Cauchy criterion directly is not very convenient. Therefore, as a rule, simpler convergence tests are used:

1) If the row
converges, then it is necessary that the common term u n tended to zero. However, this condition is not sufficient. We can only say that if the common term does not tend to zero, then the series definitely diverges. For example, the so-called harmonic series is divergent, although its common term tends to zero.

Example. Investigate the convergence of the series

We'll find
- the necessary criterion for convergence is not satisfied, which means the series diverges.

2) If a series converges, then the sequence of its partial sums is bounded.

However, this sign is also not sufficient.

For example, the series 1-1+1-1+1-1+ … +(-1) n +1 +… diverges, because the sequence of its partial sums diverges due to the fact that

However, the sequence of partial sums is limited, because
at any n.

Series with non-negative terms.

When studying series of constant sign, we will limit ourselves to considering series with non-negative terms, because simply multiplying by –1 from these series can yield series with negative terms.

Theorem. For the convergence of the series
with non-negative terms it is necessary and sufficient for the partial sums of the series to be bounded.

A sign for comparing series with non-negative terms.

Let two rows be given
And
at u n , v n  0 .

Theorem. If u n  v n at any n, then from the convergence of the series
the series converges
, and from the divergence of the series
the series diverges
.

Proof. Let us denote by S n And  n partial sums of series
And
. Because according to the conditions of the theorem, the series
converges, then its partial sums are bounded, i.e. in front of everyone n n  M, where M is a certain number. But because u n  v n, That S n   n then the partial sums of the series
are also limited, and this is sufficient for convergence.

Example. Examine the series for convergence

Because
, and the harmonic series diverges, then the series diverges
.

Example.

Because
, and the series
converges (like a decreasing geometric progression), then the series
also converges.

The following convergence sign is also used:

Theorem. If
and there is a limit
, Whereh– a number other than zero, then the series
And
behave identically in terms of convergence.

D'Alembert's sign.

(Jean Leron d'Alembert (1717 - 1783) - French mathematician)

If for a series
with positive terms there is such a numberq<1, что для всех достаточно больших ninequality holds

then a series
converges, if for all there are sufficiently largencondition is met

then a series
diverges.

D'Alembert's limiting sign.

D'Alembert's limiting criterion is a consequence of the above D'Alembert criterion.

If there is a limit
, then when < 1 ряд сходится, а при  > 1 – diverges. If = 1, then the question of convergence cannot be answered.

Example. Determine the convergence of the series .

Conclusion: the series converges.

Example. Determine the convergence of the series

Conclusion: the series converges.

Cauchy's sign. (radical sign)

If for a series
with non-negative terms there is such a numberq<1, что для всех достаточно больших ninequality holds

,

then a series
converges, if for all there are sufficiently largeninequality holds

then a series
diverges.

Consequence. If there is a limit
, then when<1 ряд сходится, а при >Row 1 diverges.

Example. Determine the convergence of the series
.

Conclusion: the series converges.

Example. Determine the convergence of the series
.

Those. The Cauchy test does not answer the question of the convergence of the series. Let us check that the necessary convergence conditions are met. As mentioned above, if a series converges, then the common term of the series tends to zero.

,

Thus, the necessary condition for convergence is not satisfied, which means the series diverges.

Integral Cauchy test.

If (x) is a continuous positive function decreasing over the interval And
then the integrals
And
behave identically in terms of convergence.

Alternating series.

Alternating rows.

An alternating series can be written as:

Where

Leibniz's sign.

If the sign of the alternating row absolute valuesu i are decreasing
and the common term tends to zero
, then the series converges.

Absolute and conditional convergence of series.

Let's consider some alternating series (with terms of arbitrary signs).

(1)

and a series composed of the absolute values ​​of the members of the series (1):

(2)

Theorem. From the convergence of series (2) follows the convergence of series (1).

Proof. Series (2) is a series with non-negative terms. If series (2) converges, then by the Cauchy criterion for any >0 there is a number N such that for n>N and any integer p>0 the following inequality is true:

According to the property of absolute values:

That is, according to the Cauchy criterion, from the convergence of series (2) the convergence of series (1) follows.

Definition. Row
called absolutely convergent, if the series converges
.

It is obvious that for series of constant sign the concepts of convergence and absolute convergence coincide.

Definition. Row
called conditionally convergent, if it converges and the series
diverges.

D'Alembert's and Cauchy's tests for alternating series.

Let
- alternating series.

D'Alembert's sign. If there is a limit
, then when<1 ряд
will be absolutely convergent, and when>

Cauchy's sign. If there is a limit
, then when<1 ряд
will be absolutely convergent, and if >1 the series will be divergent. When =1, the sign does not give an answer about the convergence of the series.

Properties of absolutely convergent series.

1) Theorem. For absolute convergence of the series
it is necessary and sufficient that it can be represented as the difference of two convergent series with non-negative terms.

Consequence. A conditionally convergent series is the difference of two divergent series with non-negative terms tending to zero.

2) In a convergent series, any grouping of the terms of the series that does not change their order preserves the convergence and magnitude of the series.

3) If a series converges absolutely, then the series obtained from it by any permutation of terms also converges absolutely and has the same sum.

By rearranging the terms of a conditionally convergent series, one can obtain a conditionally convergent series having any predetermined sum, and even a divergent series.

4) Theorem. For any grouping of members of an absolutely convergent series (in this case, the number of groups can be either finite or infinite, and the number of members in a group can be either finite or infinite), a convergent series is obtained, the sum of which is equal to the sum of the original series.

5) If the rows And converge absolutely and their sums are equal respectively S and , then a series composed of all products of the form
taken in any order, also converges absolutely and its sum is equal to S - the product of the sums of the multiplied series.

If you multiply conditionally convergent series, you can get a divergent series as a result.

Functional sequences.

Definition. If the members of the series are not numbers, but functions of X, then the series is called functional.

The study of the convergence of functional series is more complicated than the study of numerical series. The same functional series can, with the same variable values X converge, and with others - diverge. Therefore, the question of convergence of functional series comes down to determining those values ​​of the variable X, at which the series converges.

The set of such values ​​is called area of ​​convergence.

Since the limit of each function included in the convergence region of the series is a certain number, the limit of the functional sequence will be a certain function:

Definition. Subsequence ( f n (x) } converges to function f(x) on the segment if for any number >0 and any point X from the segment under consideration there is a number N = N(, x), such that the inequality

is fulfilled when n>N.

With the selected value >0, each point of the segment has its own number and, therefore, there will be an infinite number of numbers corresponding to all points of the segment. If you choose the largest of all these numbers, then this number will be suitable for all points of the segment, i.e. will be common to all points.

Definition. Subsequence ( f n (x) } converges uniformly to function f(x) on the segment , if for any number >0 there is a number N = N() such that the inequality

is fulfilled for n>N for all points of the segment.

Example. Consider the sequence

This sequence converges on the entire number line to the function f(x)=0 , because

Let's build graphs of this sequence:

sinx

As can be seen, with increasing number n the sequence graph approaches the axis X.

Functional series.

Definition. Private (partial) amounts functional range
functions are called

Definition. Functional range
called convergent at point ( x=x 0 ), if the sequence of its partial sums converges at this point. Sequence limit
called amount row
at the point X 0 .

Definition. Set of all values X, for which the series converges
called area of ​​convergence row.

Definition. Row
called uniformly convergent on the interval if the sequence of partial sums of this series converges uniformly on this interval.

Theorem. (Cauchy criterion for uniform convergence of series)

For uniform convergence of the series
it is necessary and sufficient that for any number >0 such a number existedN( ), which atn> Nand any wholep>0 inequality

would hold for all x on the interval [a, b].

Theorem. (Weierstrass test for uniform convergence)

(Karl Theodor Wilhelm Weierstrass (1815 – 1897) – German mathematician)

Row
converges uniformly and absolutely on the interval [a, b], if the moduli of its terms on the same segment do not exceed the corresponding terms of a convergent number series with positive terms:

those. there is an inequality:

.

They also say that in this case the functional series
is majorized number series
.

Example. Examine the series for convergence
.

Because
always, it is obvious that
.

Moreover, it is known that the general harmonic series when=3>1 converges, then, in accordance with the Weierstrass test, the series under study converges uniformly and, moreover, in any interval.

Example. Examine the series for convergence .

On the interval [-1,1] the inequality holds
those. according to the Weierstrass criterion, the series under study converges on this segment, but diverges on the intervals (-, -1)  (1, ).

Properties of uniformly convergent series.

1) Theorem on the continuity of the sum of a series.

If the members of the series
- continuous on the segment [a, b] function and the series converges uniformly, then its sumS(x) is a continuous function on the interval [a, b].

2) Theorem on term-by-term integration of a series.

Uniformly converging on the segment [a, b] a series with continuous terms can be integrated term by term on this interval, i.e. a series composed of integrals of its terms over the segment [a, b] , converges to the integral of the sum of the series over this segment.

3) Theorem on term-by-term differentiation of a series.

If the members of the series
converging on the segment [a, b] represent continuous functions having continuous derivatives, and a series composed of these derivatives
converges uniformly on this segment, then this series converges uniformly and can be differentiated term by term.

Based on the fact that the sum of the series is some function of the variable X, you can perform the operation of representing a function in the form of a series (expansion of a function into a series), which is widely used in integration, differentiation and other operations with functions.

In practice, the expansion of functions in power series.

Power series.

Definition. Power series is called a series of the form

.

To study the convergence of power series, it is convenient to use D'Alembert's test.

Example. Examine the series for convergence

We apply d'Alembert's sign:

.

We find that this series converges at
and diverges at
.

Now we determine the convergence at the boundary points 1 and –1.

For x = 1:
The series converges according to Leibniz's criterion (see Leibniz's sign.).

At x = -1:
the series diverges (harmonic series).

Abel's theorems.

(Nils Henrik Abel (1802 – 1829) – Norwegian mathematician)

Theorem. If the power series
converges atx = x 1 , then it converges and, moreover, for absolutely everyone
.

Proof. According to the conditions of the theorem, since the terms of the series are limited, then

Where k- some constant number. The following inequality is true:

From this inequality it is clear that when x< x 1 the numerical values ​​of the terms of our series will be less (at least not more) than the corresponding terms of the series on the right side of the inequality written above, which form a geometric progression. The denominator of this progression according to the conditions of the theorem, it is less than one, therefore, this progression is a convergent series.

Therefore, based on the comparison criterion, we conclude that the series
converges, which means the series
converges absolutely.

Thus, if the power series
converges at a point X 1 , then it converges absolutely at any point in the interval of length 2 centered at a point X = 0.

Consequence. If at x = x 1 the series diverges, then it diverges for everyone
.

Thus, for each power series there is a positive number R such that for all X such that
the series is absolutely convergent, and for all
the row diverges. In this case, the number R is called radius of convergence. The interval (-R, R) is called convergence interval.

Note that this interval can be closed on one or both sides, or not closed.

The radius of convergence can be found using the formula:

Example. Find the area of ​​convergence of the series

Finding the radius of convergence
.

Therefore, this series converges for any value X. The common term of this series tends to zero.

Theorem. If the power series
converges for a positive value x=x 1 , then it converges uniformly in any interval inside
.

Actions with power series.

INTRODUCTION

The methodological manual is intended for mathematics teachers in technical schools, as well as for second-year students of all specialties.

This paper outlines the basic concepts of series theory. Theoretical material meets the requirements of the State educational standard for secondary vocational education (Ministry of Education Russian Federation. M., 2002).

The presentation of theoretical material on the entire topic is accompanied by a consideration of a large number of examples and problems, and is conducted in accessible, as strict as possible, language. At the end of the manual there are examples and tasks that students can perform in self-control mode.

The manual is intended for part-time and full-time students.

Taking into account the level of training of technical school students, as well as the extremely limited number of hours (12 hours + 4 lbs.) allocated by the program for passing higher mathematics in technical schools, strict conclusions, which pose great difficulties for assimilation, are omitted, limiting ourselves to the consideration of examples.

BASIC CONCEPTS

Solving a problem represented in mathematical terms, for example, as a combination various functions, their derivatives and integrals, you need to be able to “bring it to a number”, which most often serves as the final answer. For this purpose, various methods have been developed in different branches of mathematics.

The branch of mathematics that allows you to solve any well-posed problem with sufficient accuracy for practical use is called series theory.

Even if some subtle concepts of mathematical analysis appeared outside of connection with the theory of series, they were immediately applied to series, which served as a tool for testing the significance of these concepts. This situation continues to this day.

Expression of the form

where ;;;…;;… are members of the series; - nth or common term of a series, is called an infinite series (series).

If the members of the series:

I. Number series

1.1. Basic concepts of number series.

A number series is a sum of the form

, (1.1)

where ,,,…,,…, called members of the series, form an infinite sequence; term is called the common term of the series.

composed of the first terms of series (1.1) are called partial sums of this series.

Each row can be associated with a sequence of partial sums .

If, with an infinite increase in number n If the partial sum of a series tends to a limit, then the series is called convergent, and the number is called the sum of a convergent series, i.e.

This entry is equivalent to

.

If the partial sum of series (1.1) with unlimited increase n does not have a finite limit (tends to or ), then such a series is called divergent .

If the row convergent , then the value for a sufficiently large n is an approximate expression for the sum of the series S.

The difference is called the remainder of the series. If a series converges, then its remainder tends to zero, i.e., and vice versa, if the remainder tends to zero, then the series converges.

1.2. Examples of number series.

Example 1. Series of the form

(1.2)

called geometric .

A geometric series is formed from the terms of a geometric progression.

It is known that the sum of its first n members Obviously: this n- th partial sum of series (1.2).

Possible cases:

Series (1.2) takes the form:

,the series diverges;

Series (1.2) takes the form:

Has no limit, the series diverges.

- a finite number, the series converges.

- the series diverges.

So, this series converges at and diverges at .

Example 2. Series of the form

(1.3)

called harmonic .

Let's write down the partial sum of this series:

The amount is greater than the amount presented as follows:

or .

If , then , or .

Therefore, if , then , i.e. the harmonic series diverges.

Example 3. Series of the form

(1.4)

called generalized harmonic .

If , then this series turns into a harmonic series, which is divergent.

If , then the terms of this series are greater than the corresponding terms of the harmonic series and, therefore, it diverges. When we have a geometric series in which ; it is convergent.

So, the generalized harmonic series converges at and diverges at .

1.3. Necessary and sufficient criteria for convergence.

A necessary sign of convergence of a series.

A series can converge only if its common term tends to zero as the number increases indefinitely: .

If , then the series diverges – this is a sufficient sign of the divergence of the series.

Sufficient signs of convergence of a series with positive terms.

A sign for comparing series with positive terms.

The series under study converges if its terms do not exceed the corresponding terms of another, obviously convergent series; the series under study diverges if its members exceed the corresponding members of another, obviously divergent series.

D'Alembert's sign.

If for a series with positive terms

the condition is satisfied, then the series converges at and diverges at .

D'Alembert's test does not give an answer if . In this case, other techniques are used to study the series.

Exercises.

Write a series based on its given common term:

Assuming ,,,…, we have an infinite sequence of numbers:

Adding its terms, we get the series

.

Doing the same, we get the series

.

Giving the values ​​1,2,3,... and taking into account that,,,..., we get the series

.

Find n- th member of the series according to its given first members:

The denominators of the terms of the series, starting from the first, are even numbers; hence, n- The th term of the series has the form .

The numerators of the members of the series form a natural series of numbers, and their corresponding denominators form a natural series of numbers, and their corresponding denominators form a natural series of numbers, starting from 3. The signs alternate according to the law or according to the law. Means, n- the th term of the series has the form . or .

Investigate the convergence of the series using the necessary convergence test and comparison test:

;

.

We find .

The necessary criterion for the convergence of a series is satisfied, but to solve the issue of convergence it is necessary to apply one of the sufficient criteria for convergence. Let's compare this series with the geometric series

,

which converges, since.

Comparing the terms of this series, starting from the second, with the corresponding terms of the geometric series, we obtain the inequalities

those. the terms of this series, starting from the second, are correspondingly smaller than the terms of the geometric series, which means that this series converges.

.

Here a sufficient criterion for the divergence of a series is satisfied; therefore, the series diverges.

We find .

The necessary criterion for the series to converge is satisfied. Let us compare this series with the generalized harmonic series

,

which converges, since, therefore, the given series also converges.

Investigate the convergence of the series using d'Alembert's test:

;

.

Substituting into the common term of the series instead n number n+ 1, we get . Let us find the limit of the ratio of the th term to n- mu member at:

Therefore, this series converges.

This means that this series diverges.

Those. the row diverges.

II. Alternating series

2.1 The concept of alternating series.

Number series

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

The number series is called signalternating , if any two standing nearby members have opposite signs.

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

;

;

.

For series with alternating signs, there is a sufficient sign of convergence (established in 1714 by Leibniz in a letter to I. Bernoulli).

2.2 Leibniz's test. Absolute and conditional convergence of series.

Theorem (Leibniz test).

An alternating series converges if:

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

The general term of the series tends to zero:.

In this case, the sum S of the series satisfies the inequalities

Notes.

Study of an alternating series of the form

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

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

The ratio allows us to obtain a simple and convenient estimate of the error that we make when 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 in modulus 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 approximately the sum of the series.

Solution: this series is of Leibniz type. It fits. You can write:

.

Taking five members, i.e. replaceable

Let's make a smaller mistake

how . So,.

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

Theorem. Let an alternating series be given

If the series converges

composed of the modules of the terms of a given series, then the alternating series itself converges.

The Leibniz convergence test for alternating series of signs serves as a sufficient criterion for the convergence of alternating series of signs.

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, but a series composed of the absolute values ​​of its terms diverges, then this series is called conditionally (not absolutely) convergent.

2.3. Exercises.

Examine for convergence (absolute or conditional) the alternating series:

And

Therefore, according to Leibniz's criterion, the series converges. Let us find out whether this series converges absolutely or conditionally.

Row , composed of the absolute values ​​of a given series, is a harmonic series that diverges. Therefore, this series converges conditionally.

The terms of this series monotonically decrease in absolute value:

, But

.

The series diverges because Leibniz's test does not hold.

Using Leibniz's test, we get

;,

those. the series converges.

.

This is a geometric series of the form where, which converges. Therefore, this series converges absolutely.

Using Leibniz's test, we have

;

, i.e. the series converges.

Let's consider a series made up of the absolute values ​​of the terms of this series:

, or

.

This is a generalized harmonic series that diverges because. Therefore, this series converges conditionally.

III. Functional range

3.1. The concept of a functional series.

A series whose members are functions of is called functional :

Giving a certain value, we get a number series

which can be either convergent or divergent.

If the resulting number series converges, then the point is called convergence point functional range; if the series diverges - divergence point functional range.

Totality numerical values arguments for which the functional series converges is called its area of ​​convergence .

In the region of convergence of a functional series, its sum is some function of:.

It is defined in the region of convergence by the equality

, Where

Partial sum of a series.

Example. Find the area of ​​convergence of the series.

Solution. This series is a series of geometric progression with denominator . Consequently, this series converges at , i.e. in front of everyone; the sum of the series is ;

, at .

3.2. Power series.

A power series is a series of the form

,

where are the numbers are called coefficients of the series , and the term is a common term of the series.

The convergence region of a power series is the set of all values ​​for which the series converges.

The number is called radius of convergence power series, if the series converges and, moreover, absolutely, and the series diverges.

Let's find the radius of convergence using d'Alembert's sign:

(does not depend on)

those. if the power series converges for any satisfying this condition and diverges for .

It follows that if there is a limit

,

then the radius of convergence of the series is equal to this limit and the power series converges at , i.e. in the interval called interval (interval) of convergence.

If , then the power series converges at a single point.

At the ends of the interval, the series can converge (absolutely or conditionally), but it can also diverge.

The convergence of a power series at and is studied using any of the convergence tests.

3.3. Exercises.

Find the area of ​​convergence of the series:

Solution. Let us find the radius of convergence of this series:

.

Consequently, this series converges absolutely on the entire number line.

Solution. Let's use d'Alembert's sign. For this series we have:

.

The series is absolutely convergent if or . Let us study the behavior of the series at the ends of the convergence interval.

When we have the series

When we have the series - this is also a convergent Leibniz series. Consequently, the region of convergence of the original series is a segment.

Solution. Let's find the radius of convergence of the series:

Consequently, the series converges at, i.e. at.

We take the series , which converges according to the Leibniz criterion.

We take a divergent series

.

Consequently, the region of convergence of the original series is the interval.

IV. Decomposition elementary functions in the Maclaurin series.

For applications, it is important to be able to expand this function into a power series, i.e. represent the function as a sum of a power series.

A Taylor series for a function is a power series of the form

If , then we get special case Taylor series

which is called near Maclaurin .

A power series within its interval of convergence can be differentiated term by term and integrated as many times as desired, and the resulting series have the same interval of convergence as the original series.

Two power series can be added and multiplied term by term according to the rules for addition and multiplication of polynomials. In this case, the convergence interval of the resulting new series coincides with the general part of the convergence intervals of the original series.

To expand a function into a Maclaurin series it is necessary:

Calculate the values ​​of the function and its successive derivatives at the point, i.e.,,,…,;

Compose a Maclaurin series by substituting the values ​​of the function and its successive derivatives into the Maclaurin series formula;

Find the interval of convergence of the resulting series using the formula

, .

Example 1. Expand the function into a Maclaurin series.

Solution. Because , then, replacing with in the expansion, we get:

Example 2. Write out the Maclaurin series of the function .

Solution. Since , then using the formula in which we replace with , we get:

,

Example 3. Expand the function in a Maclaurin series.

Solution. Let's use the formula. Because

, then replacing with we get:

, or

where, i.e. .

V. Practical tasks for self-control of students.

Using the test for comparing series, establish convergence

or divergence of series:

converges conditionally;

converges conditionally;

converges absolutely.

;

;

VII. Historical reference.

Solving many problems comes down to calculating the values ​​of functions and integrals or solving differential equations containing derivatives or differentials of unknown functions.

However, the exact execution of these mathematical operations in many cases turns out to be very difficult or impossible. In these cases, it is possible to obtain an approximate solution of many problems with any desired accuracy using series.

Series are a simple and advanced tool of mathematical analysis for approximate calculation of functions, integrals and solutions of differential equations.

And the functional row on the right.

In order to replace the “” sign with an equal sign, it is necessary to carry out some additional considerations related specifically to the infinity of the number of terms on the right side of the equality and concerning the region of convergence of the series.

When the Taylor formula takes the form in which it is called the Maclaurin formula:

Colin Maclaurin (1698 – 1746), a student of Newton, in his work “Treatise on Fluxions” (1742) established that the power series expressing an analytic function is the only one, and it will be the Taylor series generated by such a function. In the Newton binomial formula, the coefficients of the powers are the values ​​, where .

So, the ranks arose in the 18th century. as a way of representing functions that allow infinite differentiation. However, the function represented by a series was not called its sum, and in general at that time it was not yet determined what the sum of a numerical or functional series was; there were only attempts to introduce this concept.

For example, L. Euler (1707-1783), having written out the corresponding power series for a function, gave the variable a specific value. The result was a number series. Euler considered the sum of this series to be the value of the original function at the point. But this is not always true.

Scientists began to realize that a divergent series has no sum only in the 19th century, although in the 18th century. many, and above all L. Euler, worked a lot on the concepts of convergence and divergence. Euler called a series convergent if its common term tends to zero as .

In the theory of divergent series, Euler obtained many significant results, but these results did not find application for a long time. Back in 1826 N.G. Abel (1802 – 1829) called the divergent series “the devil’s invention.” Euler's results were justified only in late XIX V.

The French scientist O.L. played a major role in the formation of the concept of the sum of a convergent series. Cauchy (1789 – 1857); he did an enormous amount not only in the theory of series, but also in the theory of limits, in the development of the very concept of a limit. In 1826 Cauchy stated that a divergent series has no sum.

In 1768 French mathematician and philosopher J.L. D'Alembert investigated the ratio of the subsequent term to the previous one in a binomial series and showed that if this ratio is less than one in absolute value, then the series converges. Cauchy in 1821 proved a theorem setting out in general form a test for the convergence of positive series, now called D’Alembert’s test.

To study the convergence of alternating series, the Leibniz test is used.

G.V. Leibniz (1646 – 1716), the great German mathematician and philosopher, along with I. Newton, is the founder of differential and integral calculus.

Bibliography:

Main:

1. Bogomolov N.V., Practical lessons in mathematics. M., “Higher School”, 1990 – 495 pp.;
2. Tarasov N.P., Course of higher mathematics for technical schools. M., “Science”, 1971 – 448 pp.;
3. Zaitsev I.L., Course of higher mathematics for technical schools. M., state publishing house of technical schools - theoretical literature, 1957 - 339 pp.;
4. Pismenny D.T., Course of lectures on higher mathematics. M., “Iris Press”, 2005, part 2 – 256 p.;
5. Vygodsky M.Ya., Handbook of Higher Mathematics. M., “Science”, 1975 – 872 pp.;

Additional:

1. Gusak A.A., Higher mathematics. In 2 volumes, T.2: Textbook for university students. Mos., “TetraSystems”, 1988 – 448 p.;
2. Griguletsky V.G., Lukyanova I.V., Petunina I.A., Mathematics for students of economic specialties. Part 2. Krasnodar, 2002 – 348 pp.;
3. Griguletsky V.G. etc. Problem book in mathematics. Krasnodar. KSAU, 2003 – 170 p.;
4. Griguletsky V.G., Stepantsova K.G., Getman V.N., Tasks and exercises for students of the accounting and financial faculty. Krasnodar. 2001 – 173 pp.;
5. Griguletsky V.G., Yashchenko Z.V., Higher Mathematics. Krasnodar, 1998 – 186 pp.;
6. Malykhin V.I., Mathematics in Economics. M., “Infra-M”, 1999 – 356 p.