A sequence is a highly ordered numerical set formed according to a given law. The term "series" denotes the result of adding the terms of the corresponding sequence. For various numerical sequences, we can find the sum of all its members or the total number of elements up to a given limit.

Subsequence

This term refers to a given set of elements of the number space. Each mathematical object is given a certain formula for determining the common element of the sequence, and for most finite numerical sets there are simple formulas for determining their sum. Our program is a collection of 8 online calculators designed to calculate the sums of the most popular numerical sets. Let's start with the simplest - the natural series, which we use in everyday life to count objects.

natural sequence

When students learn numbers, the first thing they learn is to count objects, like apples. Natural numbers naturally arise when counting objects, and every child knows that 2 apples are always 2 apples, no more and no less. The natural series is given by a simple law that looks like n. The formula says that the nth member of the number set is equal to n: the first is 1, the second is 2, the four hundred and fifty-first is 451, and so on. The result of summing the first n natural numbers, that is, starting from 1, is determined by a simple formula:

∑ = 0.5n × (n+1).

Calculation of the sum of the natural series

For calculations, you will need to select the formula of the natural series n in the calculator menu and enter the number of terms in the sequence. Let's calculate the sum of the natural series from 1 to 15. By specifying n = 15, you will get the result in the form of the sequence itself:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

and the sum of the natural series equal to 120.

It is easy to check the correctness of the calculations using the above formula. For our example, the result of addition will be 0.5 × 15 × 16 = 0.5 × 240 = 120. That's right.

Sequence of squares

A quadratic sequence is formed from a natural one by squaring each term. A number of squares is formed according to the law n 2, therefore, the n-th member of the sequence will be equal to n 2: the first - 1, the second - 2 2 \u003d 4, the third - 3 2 \u003d 9 and so on. The result of summing the initial n elements of the quadratic sequence is calculated according to the law:

∑ = (n × (n+1) × (2n+1)) / 6.

With this formula, you can easily calculate the sum of the squares from 1 to n for arbitrarily large n. It is obvious that this sequence is also infinite, and as n grows, so will the total value of the numerical set.

Calculation of the sum of a square series

In this case, you will need to select the law of the square sequence n 2 in the program menu, and then select the value of n. Let's calculate the sum of the first ten terms of the sequence (n=10). The program will give the sequence itself:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

as well as an amount equal to 385.

cubic series

A row of cubes is a sequence of natural numbers cubed. The law of formation of a common element of the sequence is written as n 3 . Thus, the first member of the series is 1 3 = 1, the second is 2 3 = 8, the third is 3 3 = 27, and so on. The sum of the first n elements of the cubic series is determined by the formula:

∑ = (0.5n × (n+1)) 2

As in the previous cases, the elements of the number space tend to infinity, and the greater the number of terms, the greater the summation result.

Calculation of the sum of the cubic series

To get started, select the law of the cubic series n 3 in the calculator menu and set any value of n. Let's determine the sum of a series of 13 terms. The calculator will give us the result in the form of a sequence:

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197

and the sum of the series corresponding to it, equal to 8281.

Sequence of odd numbers

The set of natural numbers contains a subset of odd elements, that is, those that are not divisible by 2 without a remainder. The sequence of odd numbers is determined by the expression 2n - 1. According to the law, the first term of the sequence will be equal to 2 × 1 - 1 = 1, the second - 2 × 2 - 1 = 3, the third - 2 × 3 - 1 = 5 and so on. The sum of the initial n elements of an odd row is calculated using a simple formula:

Consider an example.

Calculating the sum of odd numbers

First, select the law of formation of the odd series 2n−1 in the program menu, then enter n. Let's find out the first 12 terms of the odd series and its sum. The calculator will instantly give the result as a set of numbers:

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23,

as well as the sum of the odd series, which is 144. And indeed, 12 2 = 144. That's right.

Rectangular numbers

Rectangular numbers belong to the class of curly numbers, which are a class of numerical elements needed to construct geometric shapes and bodies. For example, to build a triangle you need 3, 6 or 10 points, a square - 4, 9 or 16 points, and to lay out a tetrahedron you need 4, 10 or 20 balls or cubes. Rectangles are easy to construct using two consecutive numbers, for example, 1 and 2, 7 and 8, 56 and 57. Rectangular numbers are expressed as a product of two consecutive natural numbers. The formula for the common term of the series looks like n × (n+1). The first ten elements of such a numerical set look like:

2, 6, 12, 20, 30, 42, 56, 72, 90, 110…

With an increase in n, the value of rectangular numbers also increases, therefore, the sum of such a series will also increase.

reverse sequence

For rectangular numbers, there is an inverse sequence defined by the formula 1 / (n × (n+1)). The number set is transformed into a set of fractions and looks like this:

1/2 , 1/6, 1/12, 1/20, 1/30, 1/42, 1/56, 1/72, 1/90, 1/110…

The sum of a series of fractions is determined by the formula:

∑ = 1 - 1/(n+1).

Obviously, as the number of elements in the series increases, the value of the fraction 1/(n + 1) tends to zero, and the result of addition approaches one. Consider examples.

The sum of a rectangular series and its inverse

Let's calculate the value of a rectangular sequence for n = 20. To do this, select the law for specifying the common member of the numeric set n × (n + 1) in the online calculator menu and specify n. The program will return the instantaneous result as 3080. To calculate the inverse series, change the law to 1 / (n × (n+1)). The sum of the reciprocal numeric elements will be equal to 0.952.

Series of products of three consecutive numbers

A rectangular number set can be modified by adding another consecutive multiplier to it. Therefore, the formula for calculating the nth member of the set will be transformed into n × (n+1) × (n+2). According to this formula, the elements of a series are formed as a product of three consecutive numbers, for example, 1 × 2 × 3 or 10 × 11 × 12. The first ten elements of such a series look like:

6, 24, 60, 120, 210, 336, 504, 720, 990, 1320

This is a rapidly growing numerical set, and the sum of the corresponding series goes to infinity as n grows.

reverse sequence

As in the previous case, we can reverse the formula of the nth term and get the expression 1 / (n × (n+1) × (n+2)). Then the set of integer values ​​will be transformed into a series of fractions, the denominator of which will be the product of three consecutive numbers. The beginning of such a set looks like this:

1/6, 1/24, 1/60, 1/120, 1/210, 1/336…

The sum of the corresponding series is determined by the formula:

∑ = 0.5 × (0.5 - 1 / (n+1) × (n+2)).

Obviously, as the number of elements increases, the fraction 1 / ((n + 1) × (n + 2)) tends to zero, and the sum of the series approaches the value 0.5 × 0.5 = 0.25. Consider examples.

A series of products of three consecutive numbers and its inverse

To work with this set, you need to choose the law for determining the common element n × (n + 1) × (n + 2) and set n, for example, 100. The calculator will give you the sequence itself, as well as the value of the result of adding hundreds of numbers, equal to 26 527 650. If we choose the inverse law 1 / (n × (n + 1) × (n + 2)), the sum of a series of 100 terms will be equal to 0.250.

Conclusion

Basic concepts and definitions

Let an infinite number sequence be given:

, … (1.1)

Last year we defined a number sequence as a function of a natural argument. This means that each member of the sequence is a function of its number P: . In what follows, we will sometimes consider P equal to zero, so the numerical sequence will be defined as a function integer argument (from the words "integer").

Definition 1. Expression

(1.2)

called endless number line, or, in short, near. Sequence Members ,… are called members of a number; expression with index P- common member of the series.

It is easy to distinguish a sequence from a series: the members of the sequence are written separated by commas, the members of the series are connected by plus signs.

Thus, the concept of a series is a generalization of summation to the case of an infinite number of terms.

A series is considered given if the formula of its common term is known (given). The common term of the series (1.2) coincides with the common term of the sequence (1.1) and is also a function of the integer argument n, i.e. . For example, if a common term is given as

, (1.3)

then, putting in this formula n= 1, 2, 3,..., one can find any member of the series, and thus the entire series:

- sequence members or series members,

(1.4)

Number row.

Definition. Sum n the first members of the series is called n- oh partial sum of a series and is denoted by the symbol:

It can be written like this: .

In particular,

From all partial sums of series (1.2) we compose a numerical sequence :

(1.7)

It is called sequence of partial sums. Like any number sequence, it can have a limit, i.e. converge, or have no limit, i.e. diverge. The limit of a sequence of partial sums, if it exists, will be denoted by the letter S.

Definition. The row is called converging(row converges) if the sequence of partial sums of this series converges. At the same time, the limit S sequences of partial sums is called the sum of this series, i.e.

. (1.8)

For a convergent series with sum S, we can formally write the equality:

A series that does not have a sum (1.8) is called divergent. In particular, if , then we say that the series diverges to , and in this case we use the symbolic equality

.

Comment. It follows from equality (1.6) that any member of the series can be represented as the difference between the partial sums and :

. (1.10)

Let us represent geometrically the sequence of partial sums. In Fig. 1.1, a and b, the series converges, in Fig. 1.1, c it diverges.

Fig.1.1

Remark 3. Sometimes the number of a series member starts from zero: .

Examples of number series. Calculating the sum of a series

Example 1º.

1 + 1 + 1 + . . . + 1 + . . .

Here , .

This series diverges Þ 1 + 1 + 1 + . . . + 1 + . . .=+¥.

Example 2º .

As usual, the alternation of + and - signs is specified using the degree (-1). Here the sequence of partial sums has the form:

those. the value of the partial sum depends on the parity of the number P:

Thus, even and odd partial sums tend to two different limits:

even to zero, odd to one:

Fig.1.2

Therefore, the sequence has no limit, and the given series diverges.

Example 3º .

1 + 2 + 3 + ... + n + ...

This is an arithmetic progression with a difference. Recall that the name "arithmetic" comes from the fact that each term of this progression, starting from the second, is equal to arithmetic mean neighboring members:

.

In this progression , and the sequence of partial sums has the form:

Example 6º.

.

The output will be given below. Here, the denominator is only odd numbers.

Example 7º.

. The output will be given below.

Example 8º.

The output will be given below. The sum of the series is equal to the number e- the base of the natural logarithm.

The sum of a series is not always easy to calculate and even not always possible. Therefore, in the theory of series, a simpler problem is often solved - finding out whether the series converges or diverges. It is called the study of the convergence of the series.

FEDERAL AGENCY FOR EDUCATION

State educational institution

higher professional education

"MATI" - RUSSIAN STATE TECHNOLOGICAL UNIVERSITY IM. K.E. TSIOLKOVSKY

Department of Systems Modeling and Information Technology

Number series

Methodical instructions for practical exercises

in the discipline "Higher Mathematics"

Compilers: Egorova Yu.B.

Mamonov I.M.

Kornienko L.I.

Moscow 2005 introduction

Methodological instructions are intended for students of the day and evening department of faculty No. 14, specialties 071000, 130200, 220200.

1. Basic concepts

Let be u 1 , u 2 , u 3 , …, u n, …  an infinite numerical sequence. Expression
called endless number line, numbers u 1 , u 2 , u 3 , …, u n- members of the series;
is called the common term of the series. A series is often written in an abbreviated (folded) form:

The sum of the first n members of the number series are denoted by and call n -th partial sum of the series:

The row is called converging if it n-th partial sum with unlimited increase n tends to the final limit, i.e. if
Number called the sum of the series.

If n-th partial sum of the series at
does not tend to a finite limit, then the series is called divergent.

Example 1 Find the sum of a series
.

Decision. We have
. As:

,

Hence,

As
, then the series converges and its sum is equal to
.

2. Basic theorems on number series

Theorem 1. If the series converges
then the series converges obtained from the given series by discarding the first
members (this last row is called
-m remainder of the original row). Conversely, from the convergence
th remainder of the series implies the convergence of this series.

Theorem 2. If the series converges
and its sum is the number , then the series converges
where the sum of the last row is equal to
.

Theorem 3. If the rows converge

having sums S and Q, respectively, then the series converges, and the sum of the last series is equal to
.

Theorem 4 (A necessary criterion for the convergence of a series). If the row
converges, then
, i.e. at
the limit of the common term of the convergent series is equal to zero.

Consequence 1. If a
, then the series diverges.

Consequence 2. If a
, then it is impossible to determine the convergence or divergence of the series using the necessary criterion for convergence. A series can be either convergent or divergent.

Example 2 Investigate the convergence of the series:

Decision. Finding a common term of the series
. As:

those.
, then the series diverges (the necessary convergence condition is not satisfied).

3. Criteria for the convergence of series with positive terms

3.1. Signs of comparison

Comparison criteria are based on comparing the convergence of a given series with a series whose convergence or divergence is known. The following rows are used for comparison.

Row
composed of the terms of any decreasing geometric progression, is convergent and has the sum

Row
composed of members of an increasing geometric progression, is divergent.

Row
is divergent.

Row
is called the Dirichlet series. For >1, the Dirichlet series converges, for <1- расходится.

With =1 row
called harmonic. The harmonic series diverges.

Theorem. The first sign of comparison. Let two series with positive terms be given:

(2)

moreover, each term of series (1) does not exceed the corresponding term of series (2), i.e.,
(n= 1, 2, 3, …). Then if series (2) converges, then series (1) also converges; if series (1) diverges, then series (2) also diverges.

Comment. This criterion remains valid if the inequality
is not performed for all , but only starting from some number n= N, i.e. for all n N.

Example 3 Investigate the convergence of a series

Decision. The members of a given series are smaller than the corresponding members of the series
composed of members of an infinitely decreasing geometric progression. Since this series converges, the given series also converges.

Theorem. The second sign of comparison (the limiting form of the sign of comparison). If there is a finite non-zero limit
, then both rows and converge or diverge at the same time.

Example 4 Investigate the convergence of a series

Decision. Compare the series with the harmonic series
Find the limit of the ratio of common members of the series:

Since the harmonic series diverges, the given series also diverges.

Let a sequence of numbers R 1 , R 2 , R 3 ,…,R n ,… be given. The expression R 1 + R 2 + R 3 +…+ R n +… is called endless near, or simply near, and the numbers R 1 , R 2 , R 3 ,… - members of a number. At the same time, they mean that the accumulation of the sum of the series begins with its first members. The sum S n = is called partial sum row: for n=1 - the first partial sum, for n=2 - the second partial sum, and so on.

called convergent series, if the sequence of its partial sums has a limit, and divergent- otherwise. The concept of the sum of a series can be extended, and then some divergent series will also have sums. Exactly extended understanding amounts row will be used in the development of algorithms with the following statement of the problem: the accumulation of the sum should be performed until the next term of the series is greater in absolute value than the given value ε.

In the general case, all or part of the members of the series can be given by expressions depending on the number of the member of the series and variables. For example,

Then the question arises how to minimize the amount of calculations - to calculate the value of the next member of the series by the general formula of a member of the series(in the given example, it is represented by an expression under the sum sign), by a recursive formula (its derivation is presented below), or use recursive formulas only for parts of the expression of a series member (see below).

Derivation of a recursive formula for calculating a term of a series

Let it be required to find a series of numbers R 1 , R 2 , R 3 ,…, sequentially calculating them according to the formulas

,
, …,

To shorten the calculations in this case, it is convenient to use recurrent formula kind
, allowing to calculate the value of R N for N>1, knowing the value of the previous member of the series R N-1 , where
- an expression that can be obtained after simplifying the relation of the expression in the formula (3.1) for N to the expression for N-1:

Thus, the recursive formula will take the form
.

A comparison of the general formula for the term of the series (3.1) and the recursive one (3.2) shows that the recursive formula greatly simplifies the calculations. Let's apply it for N=2, 3 and 4 knowing that
:

Methods for calculating the value of a member of a series

To calculate the value of a series member, depending on its type, it may be preferable to use either the general formula of a series member, or a recursive formula, or mixed method of calculating the value of a member of a series, when recurrent formulas are used for one or more parts of a series member, and then their values ​​are substituted into the general formula of a series member. For example, - for a series, it is easier to calculate the value of a series member
according to its general formula
(compare with
- recurrent formula); - for a row
it is better to use the recursive formula
; - for a series, a mixed method should be applied, calculating A N \u003d X 3N using the recursive formula
, N=2, 3,… with A 1 =1 and B N =N! - also by the recursive formula
, N=2, 3,… at B 1 =1, and then - a member of the series
- according to the general formula, which will take the form
.

Example 3.2.1 task execution

Calculate with accuracy ε for 0 o  X  45 o

using a recursive formula to calculate a term of a series:

,

the exact value of the cos X function,

absolute and relative errors of the approximate value.

program Project1;

($APPTYPE CONSOLE)

K=Pi/180; //Factor to convert from degrees to radians

Eps: Extended=1E-8;

X: Extended=15;

R, S, Y, D: Extended;

($IFNDEF DBG) //Statements not used for debugging

Write("Enter the required precision: ");

Write("Enter the angle value in degrees: ");

D:=Sqr(K*X); // Convert X to radians and squaring

//Set initial values ​​for variables

//Loop for calculating the members of the series and accumulating their sum.

//Execute while the modulus of the next member of the series is greater than Eps.

while Abs(R)>Eps do

if N<10 then //Вывод, используемый при отладке

WriteLn("N=", N, " R=", R:14:11, " S=", S:14:11);

//Output of calculation results:

WriteLn(N:14," = Number of steps reached",

"specified accuracy");

WriteLn(S:14:11," = Approximate function value");

WriteLn(Cos(K*X):14:11," = Exact function value");

WriteLn(Abs(Cos(K*X)-S):14:11," = Absolute error");

WriteLn(Abs((Cos(K*X)-S)/Cos(K*X)):14:11,

" = relative error");

Find the sum of a series of numbers. If it is not possible to find it, then the system calculates the sum of the series with a certain accuracy.

Series Convergence

This calculator can determine whether a series converges, and also shows which signs of convergence work and which do not.

He also knows how to determine the convergence of power series.

A series graph is also built, where you can see the rate of convergence of the series (or divergence).

Rules for entering expressions and functions

Expressions can consist of functions (notations are given in alphabetical order): absolute(x) Absolute value x
(module x or |x|) arccos(x) Function - arc cosine of x arccosh(x) Arc cosine hyperbolic from x arcsin(x) Arcsine from x arcsinh(x) Arcsine hyperbolic from x arctg(x) Function - arc tangent from x arctgh(x) The arc tangent is hyperbolic from x e e a number that is approximately equal to 2.7 exp(x) Function - exponent from x(which is e^x) log(x) or log(x) Natural logarithm of x
(To obtain log7(x), you need to enter log(x)/log(7) (or, for example, for log10(x)=log(x)/log(10)) pi The number is "Pi", which is approximately equal to 3.14 sin(x) Function - Sine of x cos(x) Function - Cosine of x sinh(x) Function - Hyperbolic sine of x cash(x) Function - Hyperbolic cosine of x sqrt(x) The function is the square root of x sqr(x) or x^2 Function - Square x tg(x) Function - Tangent from x tgh(x) Function - Hyperbolic tangent of x cbrt(x) The function is the cube root of x floor(x) Function - rounding x down (example floor(4.5)==4.0) sign(x) Function - Sign x erf(x) Error function (Laplace or probability integral)

You can use the following operations in expressions: Real numbers enter in the form 7.5 , not 7,5 2*x- multiplication 3/x- division x^3- exponentiation x + 7- addition x - 6- subtraction