Geometric progression. Comprehensive guide with examples (2019)
Number sequence
So, let's sit down and start writing some numbers. For example:
You can write any numbers, and there can be as many of them as you like (in our case, there are them). No matter how many numbers we write, we can always say which one is first, which one is second, and so on until the last, that is, we can number them. This is an example of a number sequence:
Number sequence is a set of numbers, each of which can be assigned a unique number.
For example, for our sequence:
The assigned number is specific to only one number in the sequence. In other words, there are no three second numbers in the sequence. The second number (like the th number) is always the same.
The number with the number is called the nth member of the sequence.
We usually call the entire sequence by some letter (for example,), and each member of this sequence is the same letter with an index equal to the number of this member: .
In our case:
The most common types of progression are arithmetic and geometric. In this topic we will talk about the second type - geometric progression.
Why is geometric progression needed and its history?
Even in ancient times, the Italian mathematician monk Leonardo of Pisa (better known as Fibonacci) dealt with the practical needs of trade. The monk was faced with the task of determining with the help of which least amount weights can you weigh the goods? In his works, Fibonacci proves that such a system of weights is optimal: This is one of the first situations in which people had to deal with a geometric progression, which you have probably already heard about and have at least general concept. Once you fully understand the topic, think about why such a system is optimal?
Currently, in life practice, geometric progression It manifests itself when investing money in a bank, when the amount of interest is calculated on the amount accumulated in the account for the previous period. In other words, if you put money on a time deposit in a savings bank, then after a year the deposit will increase by the original amount, i.e. new amount will be equal to the contribution multiplied by. In another year, this amount will increase by, i.e. the amount obtained at that time will again be multiplied by and so on. Similar situation described in problems for calculating the so-called compound interest- the percentage is taken each time from the amount that is in the account, taking into account previous interest. We'll talk about these tasks a little later.
There are many more simple cases, where geometric progression is applied. For example, the spread of influenza: one person infected another person, they, in turn, infected another person, and thus the second wave of infection is a person, and they, in turn, infected another... and so on...
By the way, a financial pyramid, the same MMM, is a simple and dry calculation based on the properties of a geometric progression. Interesting? Let's figure it out.
Geometric progression.
Let's say we have number sequence:
You will immediately answer that this is easy and the name of such a sequence is an arithmetic progression with the difference of its terms. How about this:
If you subtract the previous one from the subsequent number, you will see that each time you get new difference(etc.), but the sequence definitely exists and is easy to notice - each next number times more than the previous one!
This type of number sequence is called geometric progression and is designated.
Geometric progression () is a numerical sequence, the first term of which is different from zero, and each term, starting from the second, is equal to the previous one, multiplied by the same number. This number is called the denominator of a geometric progression.
The restrictions that the first term ( ) is not equal and are not random. Let's assume that there are none, and the first term is still equal, and q is equal to, hmm.. let it be, then it turns out:
Agree that this is no longer a progression.
As you understand, we will get the same results if there is any number other than zero, a. In these cases, there will simply be no progression, since the entire number series there will either be all zeros, or one number and all the rest zeros.
Now let's talk in more detail about the denominator of the geometric progression, that is, o.
Let's repeat: - this is the number how many times does each subsequent term change? geometric progression.
What do you think it could be? That's right, positive and negative, but not zero (we talked about this a little higher).
Let's assume that ours is positive. Let in our case, a. What is the value of the second term and? You can easily answer that:
That's right. Accordingly, if, then all subsequent terms of the progression have same sign- They are positive.
What if it's negative? For example, a. What is the value of the second term and?
This is a completely different story
Try to count the terms of this progression. How much did you get? I have. Thus, if, then the signs of the terms of the geometric progression alternate. That is, if you see a progression with alternating signs for its members, then its denominator is negative. This knowledge can help you test yourself when solving problems on this topic.
Now let's practice a little: try to determine which number sequences are a geometric progression and which are an arithmetic progression:
Got it? Let's compare our answers:
- Geometric progression - 3, 6.
- Arithmetic progression - 2, 4.
- It is neither an arithmetic nor a geometric progression - 1, 5, 7.
Let's return to our last progression and try to find its member, just like in the arithmetic one. As you may have guessed, there are two ways to find it.
We successively multiply each term by.
So, the th term of the described geometric progression is equal to.
As you already guessed, now you yourself will derive a formula that will help you find any member of the geometric progression. Or have you already developed it for yourself, describing how to find the th member step by step? If so, then check the correctness of your reasoning.
Let us illustrate this with the example of finding the th term of this progression:
In other words:
Find the value of the term of the given geometric progression yourself.
Happened? Let's compare our answers:
Please note that you got exactly the same number as in the previous method, when we sequentially multiplied by each previous term of the geometric progression.
Let's try to "depersonalize" this formula- Let's put it in general form and get:
The derived formula is true for all values - both positive and negative. Check this yourself by calculating the terms of the geometric progression with the following conditions: , A.
Did you count? Let's compare the results:
Agree that it would be possible to find a term of a progression in the same way as a term, however, there is a possibility of calculating incorrectly. And if we have already found the th term of the geometric progression, then what could be simpler than using the “truncated” part of the formula.
Infinitely decreasing geometric progression.
More recently, we talked about what can be either greater or less than zero, however, there is special meanings for which the geometric progression is called infinitely decreasing.
Why do you think this name is given?
First, let's write down some geometric progression consisting of terms.
Let's say, then:
We see that each subsequent term is less than the previous one by a factor, but will there be any number? You will immediately answer - “no”. That is why it is infinitely decreasing - it decreases and decreases, but never becomes zero.
To clearly understand how this looks visually, let's try to draw a graph of our progression. So, for our case, the formula takes the following form:
On graphs we are accustomed to plotting dependence on, therefore:
The essence of the expression has not changed: in the first entry we showed the dependence of the value of a member of a geometric progression on its ordinal number, and in the second entry we simply took the value of a member of a geometric progression as, and designated the ordinal number not as, but as. All that remains to be done is to build a graph.
Let's see what you got. Here's the graph I came up with:
Do you see? The function decreases, tends to zero, but never crosses it, so it is infinitely decreasing. Let’s mark our points on the graph, and at the same time what the coordinate and means:
Try to schematically depict a graph of a geometric progression if its first term is also equal. Analyze what is the difference with our previous graph?
Did you manage? Here's the graph I came up with:
Now that you have fully understood the basics of the topic of geometric progression: you know what it is, you know how to find its term, and you also know what an infinitely decreasing geometric progression is, let's move on to its main property.
Property of geometric progression.
Do you remember the property of members arithmetic progression? Yes, yes, how to find the value a certain number progression, when there are previous and subsequent values of the members of this progression. Do you remember? This:
Now we are faced with exactly the same question for the terms of a geometric progression. To withdraw a similar formula, let's start drawing and reasoning. You'll see, it's very easy, and if you forget, you can get it out yourself.
Let's take another simple geometric progression, in which we know and. How to find? With arithmetic progression it is easy and simple, but what about here? In fact, there is nothing complicated in geometric either - you just need to write down each value given to us according to the formula.
You may ask, what should we do about it now? Yes, very simple. First, let's depict these formulas in a picture and try to do various manipulations with them in order to arrive at the value.
Let's abstract from the numbers that are given to us, let's focus only on their expression through the formula. We need to find the value highlighted orange, knowing the members adjacent to it. Let's try to produce with them various actions, as a result of which we can get.
Addition.
Let's try to add two expressions and we get:
From given expression, as you see, we cannot express it in any way, therefore, we will try another option - subtraction.
Subtraction.
As you can see, we cannot express this either, therefore, let’s try to multiply these expressions by each other.
Multiplication.
Now look carefully at what we have by multiplying the terms of the geometric progression given to us in comparison with what needs to be found:
Guess what I'm talking about? That's right, to find we need to take Square root from the geometric progression numbers adjacent to the desired one multiplied by each other:
Here you go. You yourself derived the property of geometric progression. Try writing this formula in general view. Happened?
Forgot the condition for? Think about why it is important, for example, try to calculate it yourself. What will happen in this case? That's right, complete nonsense because the formula looks like this:
Accordingly, do not forget this limitation.
Now let's calculate what it equals
Correct answer - ! If you didn't forget the second one when calculating possible meaning, then you are a great fellow and can immediately move on to training, and if you forgot, read what is discussed below and pay attention to why it is necessary to write down both roots in the answer.
Let's draw both of our geometric progressions - one with a value and the other with a value and check whether both of them have the right to exist:
In order to check whether such a geometric progression exists or not, it is necessary to see whether it is the same between all given members? Calculate q for the first and second cases.
See why we have to write two answers? Because the sign of the term you are looking for depends on whether it is positive or negative! And since we don’t know what it is, we need to write both answers with a plus and a minus.
Now that you have mastered the main points and derived the formula for the property of geometric progression, find, knowing and
Compare your answers with the correct ones:
What do you think, what if we were given not the values of the terms of the geometric progression adjacent to the desired number, but equidistant from it. For example, we need to find, and given and. Can we use the formula we derived in this case? Try to confirm or refute this possibility in the same way, describing what each value consists of, as you did when you originally derived the formula, at.
What did you get?
Now look carefully again.
and correspondingly:
From this we can conclude that the formula works not only with neighboring with the desired terms of the geometric progression, but also with equidistant from what the members are looking for.
Thus, our initial formula takes the form:
That is, if in the first case we said that, now we say that it can be equal to any natural number, which is smaller. The main thing is that it is the same for both given numbers.
Practice on specific examples, just be extremely careful!
- , . Find.
- , . Find.
- , . Find.
Decided? I hope you were extremely attentive and noticed a small catch.
Let's compare the results.
In the first two cases, we calmly apply the above formula and get the following values:
In the third case, upon closer examination serial numbers numbers given to us, we understand that they are not equidistant from the number we are looking for: is the previous date, but is removed at the position, so it is not possible to apply the formula.
How to solve it? It's actually not as difficult as it seems! Let us write down what each number given to us and the number we are looking for consists of.
So we have and. Let's see what we can do with them? I suggest dividing by. We get:
We substitute our data into the formula:
The next step we can find - for this we need to take cube root from the resulting number.
Now let's look again at what we have. We have it, but we need to find it, and it, in turn, is equal to:
We found all the necessary data for the calculation. Substitute into the formula:
Our answer: .
Try solving another similar problem yourself:
Given: ,
Find:
How much did you get? I have - .
As you can see, essentially you need remember just one formula- . You can withdraw all the rest yourself without any difficulty at any time. To do this, simply write the simplest geometric progression on a piece of paper and write down what each of its numbers is equal to, according to the formula described above.
The sum of the terms of a geometric progression.
Now let's look at formulas that allow us to quickly calculate the sum of terms of a geometric progression in a given interval:
To derive the formula for the sum of terms of a finite geometric progression, multiply all parts of the above equation by. We get:
Look carefully: what do the last two formulas have in common? That's right, common members, for example, and so on, except for the first and last member. Let's try to subtract the 1st from the 2nd equation. What did you get?
Now express the term of the geometric progression through the formula and substitute the resulting expression into our last formula:
Group the expression. You should get:
All that remains to be done is to express:
Accordingly, in this case.
What if? What formula works then? Imagine a geometric progression at. What is she like? Correct row identical numbers, accordingly the formula will look like in the following way:
There are many legends about both arithmetic and geometric progression. One of them is the legend of Set, the creator of chess.
Many people know that the game of chess was invented in India. When the Hindu king met her, he was delighted with her wit and the variety of positions possible in her. Having learned that it was invented by one of his subjects, the king decided to personally reward him. He summoned the inventor to himself and ordered him to ask him for everything he wanted, promising to fulfill even the most skillful desire.
Seta asked for time to think, and when the next day Seta appeared before the king, he surprised the king with the unprecedented modesty of his request. He asked to give a grain of wheat for the first square of the chessboard, a grain of wheat for the second, a grain of wheat for the third, a fourth, etc.
The king was angry and drove Seth away, saying that the servant's request was unworthy of the king's generosity, but promised that the servant would receive his grains for all the squares of the board.
And now the question: using the formula for the sum of the terms of a geometric progression, calculate how many grains Seth should receive?
Let's start reasoning. Since, according to the condition, Seth asked for a grain of wheat for the first square of the chessboard, for the second, for the third, for the fourth, etc., then we see that in the problem we're talking about about geometric progression. What does it equal in this case?
Right.
Total squares of the chessboard. Respectively, . We have all the data, all that remains is to plug it into the formula and calculate.
To imagine at least approximately the “scale” given number, transform using the properties of the degree:
Of course, if you want, you can take a calculator and calculate what number you end up with, and if not, you’ll have to take my word for it: the final value of the expression will be.
That is:
quintillion quadrillion trillion billion million thousand.
Phew) If you want to imagine the enormity of this number, then estimate how large a barn would be required to accommodate the entire amount of grain.
If the barn is m high and m wide, its length would have to extend for km, i.e. twice as far as from the Earth to the Sun.
If the king were strong in mathematics, he could have invited the scientist himself to count the grains, because to count a million grains, he would need at least a day of tireless counting, and given that it is necessary to count quintillions, the grains would have to be counted throughout his life.
Now let’s solve a simple problem involving the sum of terms of a geometric progression.
A student of class 5A Vasya fell ill with the flu, but continues to go to school. Every day Vasya infects two people, who, in turn, infect two more people, and so on. There are only people in the class. In how many days will the whole class be sick with the flu?
So, the first term of the geometric progression is Vasya, that is, a person. The th term of the geometric progression is the two people he infected on the first day of his arrival. total amount members of the progression is equal to the number of students in 5A. Accordingly, we talk about a progression in which:
Let's substitute our data into the formula for the sum of the terms of a geometric progression:
The whole class will get sick within days. Don't believe formulas and numbers? Try to portray the “infection” of students yourself. Happened? Look how it looks for me:
Calculate for yourself how many days it would take for students to get sick with the flu if each one infected a person, and there were only one person in the class.
What value did you get? It turned out that everyone started getting sick after a day.
As you see, similar task and the drawing to it resembles a pyramid, in which each subsequent one “brings” new people. However, sooner or later a moment comes when the latter cannot attract anyone. In our case, if we imagine that the class is isolated, the person from closes the chain (). Thus, if a person were involved in financial pyramid, in which money was given if you bring two other participants, then the person (or general case) would not have brought anyone, and therefore would have lost everything they invested in this financial scam.
Everything that was said above refers to a decreasing or increasing geometric progression, but, as you remember, we have special kind- an infinitely decreasing geometric progression. How to calculate the sum of its members? And why does this type of progression have certain features? Let's figure it out together.
So, first, let's look again at this drawing of an infinitely decreasing geometric progression from our example:
Now let’s look at the formula for the sum of a geometric progression, derived a little earlier:
or
What are we striving for? That's right, the graph shows that it tends to zero. That is, at, will be almost equal, respectively, when calculating the expression we will get almost. In this regard, we believe that when calculating the sum of an infinitely decreasing geometric progression, this bracket can be neglected, since it will be equal.
- formula is the sum of the terms of an infinitely decreasing geometric progression.
IMPORTANT! We use the formula for the sum of terms of an infinitely decreasing geometric progression only if the condition explicitly states that we need to find the sum infinite number of members.
If a specific number n is specified, then we use the formula for the sum of n terms, even if or.
Now let's practice.
- Find the sum of the first terms of the geometric progression with and.
- Find the sum of the terms of an infinitely decreasing geometric progression with and.
I hope you were extremely careful. Let's compare our answers:
Now you know everything about geometric progression, and it’s time to move from theory to practice. The most common geometric progression problems encountered on the exam are problems calculating compound interest. These are the ones we will talk about.
Problems on calculating compound interest.
You've probably heard of the so-called compound interest formula. Do you understand what it means? If not, let’s figure it out, because once you understand the process itself, you will immediately understand what geometric progression has to do with it.
We all go to the bank and know that there are different conditions on deposits: this is the term, and additional service, and interest with two different ways its calculations - simple and complex.
WITH simple interest everything is more or less clear: interest is accrued once at the end of the deposit term. That is, if we say that we deposit 100 rubles for a year, then they will be credited only at the end of the year. Accordingly, by the end of the deposit we will receive rubles.
Compound interest- this is an option in which it occurs interest capitalization, i.e. their addition to the deposit amount and subsequent calculation of income not from the initial, but from the accumulated deposit amount. Capitalization does not occur constantly, but with some frequency. As a rule, such periods are equal and most often banks use a month, quarter or year.
Let’s assume that we deposit the same rubles annually, but with monthly capitalization of the deposit. What are we doing?
Do you understand everything here? If not, let's figure it out step by step.
We brought rubles to the bank. By the end of the month, we should have an amount in our account consisting of our rubles plus interest on them, that is:
Agree?
We can take it out of brackets and then we get:
Agree, this formula is already more similar to what we wrote at the beginning. All that's left is to figure out the percentages
In the problem statement we are told about annual rates. As you know, we do not multiply by - we convert percentages to decimal fractions, that is:
Right? Now you may ask, where did the number come from? Very simple!
I repeat: the problem statement says about ANNUAL interest that accrues MONTHLY. As you know, in a year of months, accordingly, the bank will charge us a portion of the annual interest per month:
Realized it? Now try to write what this part of the formula would look like if I said that interest is calculated daily.
Did you manage? Let's compare the results:
Well done! Let's return to our task: write how much will be credited to our account in the second month, taking into account that interest is accrued on the accumulated deposit amount.
Here's what I got:
Or, in other words:
I think that you have already noticed a pattern and saw a geometric progression in all this. Write what its member will be equal to, or, in other words, what amount of money we will receive at the end of the month.
Did? Let's check!
As you can see, if you put money in a bank for a year at a simple interest rate, you will receive rubles, and if at a compound interest rate, you will receive rubles. The benefit is small, but this only happens during the th year, but for more a long period capitalization is much more profitable:
Let's consider another type of problem: compound interest. After what you have figured out, it will be elementary for you. So, the task:
The Zvezda company began investing in the industry in 2000, with capital in dollars. Every year since 2001, it has received a profit that is equal to the previous year's capital. How much profit will the Zvezda company receive at the end of 2003 if profits were not withdrawn from circulation?
Capital of the Zvezda company in 2000.
- capital of the Zvezda company in 2001.
- capital of the Zvezda company in 2002.
- capital of the Zvezda company in 2003.
Or we can write briefly:
For our case:
2000, 2001, 2002 and 2003.
Respectively:
rubles
Please note that in this problem we do not have a division either by or by, since the percentage is given ANNUALLY and it is calculated ANNUALLY. That is, when reading a problem on compound interest, pay attention to what percentage is given and in what period it is calculated, and only then proceed to calculations.
Now you know everything about geometric progression.
Training.
- Find the term of the geometric progression if it is known that, and
- Find the sum of the first terms of the geometric progression if it is known that, and
- The MDM Capital company began investing in the industry in 2003, with capital in dollars. Every year since 2004, it has received a profit that is equal to the previous year's capital. MSK company Cash flows"began investing in the industry in 2005 in the amount of $10,000, starting to make a profit in 2006 in the amount of. By how many dollars is the capital of one company greater than the other at the end of 2007, if profits were not withdrawn from circulation?
Answers:
- Since the problem statement does not say that the progression is infinite and you need to find the sum specific number its members, then the calculation is carried out according to the formula:
MDM Capital Company:2003, 2004, 2005, 2006, 2007.
- increases by 100%, that is, 2 times.
Respectively:
rubles
MSK Cash Flows company:2005, 2006, 2007.
- increases by, that is, by times.
Respectively:
rubles
rubles
Let's summarize.
1) Geometric progression ( ) is a numerical sequence, the first term of which is different from zero, and each term, starting from the second, is equal to the previous one, multiplied by the same number. This number is called the denominator of a geometric progression.
2) The equation of the terms of the geometric progression is .
3) can take any values except and.
- if, then all subsequent terms of the progression have the same sign - they are positive;
- if, then all subsequent terms of the progression alternate signs;
- when - the progression is called infinitely decreasing.
4) , with - property of geometric progression (adjacent terms)
or
, at (equidistant terms)
When you find it, don’t forget that there should be two answers.
For example,
5) The sum of the terms of the geometric progression is calculated by the formula:
or
If the progression is infinitely decreasing, then:
or
IMPORTANT! We use the formula for the sum of terms of an infinitely decreasing geometric progression only if the condition explicitly states that we need to find the sum infinite number members.
6) Problems involving compound interest are also calculated using the formula for the th term of a geometric progression, provided that cash were not withdrawn from circulation:
GEOMETRIC PROGRESSION. BRIEFLY ABOUT THE MAIN THINGS
Geometric progression( ) is a numerical sequence, the first term of which is different from zero, and each term, starting from the second, is equal to the previous one, multiplied by the same number. This number is called denominator of a geometric progression.
Denominator of geometric progression can take any value except and.
- If, then all subsequent terms of the progression have the same sign - they are positive;
- if, then all subsequent members of the progression alternate signs;
- when - the progression is called infinitely decreasing.
Equation of terms of geometric progression - .
Sum of terms of a geometric progression calculated by the formula:
or
NUMERIC SEQUENCES VI
§ l48. Sum of an infinitely decreasing geometric progression
Until now, when talking about sums, we have always assumed that the number of terms in these sums is finite (for example, 2, 15, 1000, etc.). But when solving some problems (especially higher mathematics) one has to deal with the sums of an infinite number of terms
S= a 1 + a 2 + ... + a n + ... . (1)
What are these amounts? A-priory the sum of an infinite number of terms a 1 , a 2 , ..., a n , ... is called the limit of the sum S n first P numbers when P -> ∞ :
S=S n = (a 1 + a 2 + ... + a n ). (2)
Limit (2), of course, may or may not exist. Accordingly, they say that the sum (1) exists or does not exist.
How can we find out whether the sum (1) exists in each specific case? Common decision This issue goes far beyond the scope of our program. However, there is one important special case, which we now have to consider. We will talk about summing the terms of an infinitely decreasing geometric progression.
Let a 1 , a 1 q , a 1 q 2, ... is an infinitely decreasing geometric progression. This means that | q |< 1. Сумма первых P terms of this progression is equal
From the main theorems about limits variables(see § 136) we get:
But 1 = 1, a qn = 0. Therefore
So, the sum of an infinitely decreasing geometric progression is equal to the first term of this progression divided by one minus the denominator of this progression.
1) The sum of the geometric progression 1, 1/3, 1/9, 1/27, ... is equal to
and the sum of the geometric progression is 12; -6; 3; - 3 / 2 , ... equal
2) Simple periodic fraction 0.454545 ... convert to ordinary.
To solve this problem, let's imagine given fraction as an infinite sum:
The right side of this equality is the sum of an infinitely decreasing geometric progression, the first term of which is equal to 45/100, and the denominator is 1/100. That's why
Using the described method, a general rule for converting simple periodic fractions into ordinary fractions can be obtained (see Chapter II, § 38):
To convert a simple periodic fraction into an ordinary fraction, you need to do the following: put the period in the numerator decimal, and the denominator is a number consisting of nines taken as many times as there are digits in the period of the decimal fraction.
3) Convert the mixed periodic fraction 0.58333 .... into an ordinary fraction.
Let's imagine this fraction as an infinite sum:
On the right side of this equality, all terms, starting from 3/1000, form an infinitely decreasing geometric progression, the first term of which is equal to 3/1000, and the denominator is 1/10. That's why
Using the described method, a general rule for converting mixed periodic fractions into ordinary fractions can be obtained (see Chapter II, § 38). We deliberately do not present it here. There is no need to remember this cumbersome rule. It is much more useful to know that any mixed periodic fraction can be represented as the sum of an infinitely decreasing geometric progression and a certain number. And the formula
for the sum of an infinitely decreasing geometric progression, you must, of course, remember.
As an exercise, we suggest that you, in addition to the problems No. 995-1000 given below, once again turn to problem No. 301 § 38.
Exercises
995. What is called the sum of an infinitely decreasing geometric progression?
996. Find the sums of infinitely decreasing geometric progressions:
997. At what values X progression
is it infinitely decreasing? Find the sum of such a progression.
998.V equilateral triangle with the side A inscribed by joining the midpoints of its sides new triangle; a new triangle is inscribed in this triangle in the same way, and so on ad infinitum.
a) the sum of the perimeters of all these triangles;
b) the sum of their areas.
999. Square with side A inscribed by joining the midpoints of its sides new square; a square is inscribed in this square in the same way, and so on ad infinitum. Find the sum of the perimeters of all these squares and the sum of their areas.
1000. Compose an infinitely decreasing geometric progression such that its sum is equal to 25/4, and the sum of the squares of its terms is equal to 625/24.
By introducing the notation at the beginning of the chapter, we cleverly evaded the question of infinite sums by essentially saying, “Let’s leave that for later. In the meantime, we can assume that all occurring sums have only a finite number of non-zero terms! But the time of reckoning has finally come - we must face the fact that
the amounts can be infinite. And to tell the truth, infinite amounts accompanied by both pleasant and unpleasant circumstances.
First, about the unpleasant: it turns out that the methods that we used when handling sums are not always valid for infinite sums. And now for the good stuff: there is a vast arranged class endless amounts, for which all the operations that we performed were completely legal. The reasons behind both circumstances will become clear after we find out true meaning summation.
Everyone knows what it is final amount: we add all the terms to the total, one after another, until they all add up. But an infinite amount should be determined more delicately so as not to get into trouble.
is equal to 2, since when we double it we get
But then, following the same logic, we would need to calculate the amount
equal to -1, because when we double it we get
Something strange is happening: how can you get a negative number, summing up positive values? It seems better to leave the sum of T undefined, and perhaps we should assume that since the terms in T become greater than any fixed finite number. (Note that the quantity is another “solution” to the equation; it also “solves” the equation
Let's try to give a proper definition of the value of an arbitrary sum where the set K can be infinite. To begin with, assume that all terms of a are non-negative. In this case appropriate definition is not difficult to find: if for any finite subset there is a limiting constant A such that
then we take the sum to be the smallest of all such A. (As follows from the well-known properties of real numbers, the set of all such A always contains the smallest element.) But if such a limiting constant A does not exist, we take this to mean that if A -
some real number, then there is some finite number of terms of a, the sum of which exceeds A.
The definition in the previous paragraph is formulated so delicately that it does not depend on any order that may exist in the index set K. Therefore, the arguments that we are going to give will be valid not only for sums over a set of integers, but also for multiple sums with many indexes
In particular, when K is the set of non-negative integers, our definition for the non-negative terms a means that
And here's why: any non-decreasing sequence of real numbers has a limit (possibly equal to If this limit is equal, some finite set of non-negative integers, all of which are then; therefore, either or A is a limiting constant. But if A is some number less established border A, then there is such that, in addition, a finite set testifies to the fact that A is not a limiting constant.
Now you can easily calculate the magnitudes of specific infinite sums in accordance with the definition just given. For example, if then
In particular, the infinite sums and T, which were discussed a moment ago, are equal to 2 and, respectively, as we expected. Another noteworthy example:
Now let's consider the case when, along with non-negative sums, the sum can contain negative terms. What, for example, should be the amount of
If we group the terms in pairs, we get:
so the amount turns out to be equal to zero; but if we start grouping into pairs a step later, we get
i.e. the sum is equal to one.
We could also try to put in the formula since we know that this formula is valid for but then we will be forced to admit that this infinite sum is equal because it is the sum of integers!
Another interesting example is the sum infinite in both directions in which at k 0 and at E can be written as
If we calculate this sum by starting from the “central” element and moving outward,
then we get 1; and we get the same 1 if we move all the brackets one element to the left,
since the sum of all numbers enclosed in inner brackets is
Similar reasoning shows that the value of the sum remains equal to 1 if these brackets are moved any fixed number of elements to the left or to the right - this strengthens our opinion that the sum really equals 1. But, on the other hand, if we group the terms as follows:
then the pair of inner brackets will contain numbers
In ch. 9 it will be shown that, therefore, this method grouping leads to the idea that a sum that is infinite in both directions should actually be equal to
There is something meaningless about the amount that gives different meanings when adding its members different ways. Modern analysis manuals include whole line definitions by which meaningful meanings are assigned to such pathological sums; but if we borrow these definitions, we will not be able to operate with the -notation as freely as we have done so far. The purposes of this book are such that we do not need refined clarifications of the concept " conditional convergence" - we will adhere to such a definition of infinite sums, which leaves in force all the operations we used in this chapter.
In essence, our definition of infinite sums is quite simple. Let K be a set and let a be a real-valued term of the sum defined for each . (In fact, it can mean several indices so that the set K itself can be multidimensional.) Any real number x can be represented as the difference of its positive and negative parts,
(Either or We have already explained how to determine the magnitude of infinite sums since they are non-negative. Therefore, our general definition is this:
unless both sums on the right side are equal. IN the latter case Hlek's amount remains uncertain.
Let Tskekak and If the sums are finite, then they say that the sum converges absolutely to . If it is finite, then they say that the sum diverges to Similarly, if it is finite, then they say that it diverges to If, then they say nothing.
We started with a definition that “worked” for non-negative terms of the sum, and then extended it to any real-valued terms. If the members of the sum are complex numbers, then our definition can obviously be extended to this case: the sum is defined as - real and imaginary part a, provided that both of these sums exist. otherwise the amount of Hkek is not determined. (See exercise 18.)
The unfortunate thing, as already mentioned, is that some infinite amounts have to be left undefined because the operations we perform with them can lead to absurdities. (See exercise 34.) The nice thing is that all the operations from this chapter are absolutely valid whenever we are dealing with sums that absolutely converge in the sense just established.
We can confirm this pleasant fact by demonstrating that each of our sum transformation rules leaves the magnitude of any absolutely convergent sum unchanged. More specifically, this means that one should check the fulfillment of the distributive, combinational and commutative laws, plus the rule according to which one can begin to sum over any variable; everything else we did in this chapter can be derived from these four basic sum operations.
The distribution law (2.15) can be formulated more strictly as follows: if the sum Xek a converges absolutely to and if c is some complex number, then Lkek absolutely converges to This can be proven by first dividing the sum into real and imaginary, then into positive and negative parts, as they did before, and proving the special case when each term of the sum is non-negative. The proof in this particular case works due to the fact that for any finite set the last fact can be proved by induction on the size of the set
Combination law(2.16) can be formulated as follows: if the sums converge absolutely to A and B, respectively, then the sum absolutely converges to It turns out that this is a special case of more general theorem, which we will prove shortly.
There is actually no need to prove the commutative law (2.17), since when discussing formula (2.35) we showed how to derive it as a special case general rule changes in the order of summation.
Definitions and properties of infinitesimal and infinitely large functions at a point. Proofs of properties and theorems. Relationship between infinitesimal and infinitely large functions.
Definitions of infinitesimal and infinitesimal functions
Let x 0 is a finite or infinite point: ∞, -∞ or +∞.
Definition of an infinitesimal function
Function α (x) called infinitesimal as x tends to x 0
0
, and it is equal to zero:
.
Definition of an infinitely large function
Function f (x) called infinitely large as x tends to x 0
, if the function has a limit as x → x 0
, and it is equal to infinity:
.
Properties of infinitesimal functions
Property of the sum, difference and product of infinitesimal functions
Sum, difference and product finite number of infinitesimal functions as x → x 0 is an infinitesimal function as x → x 0 .
This property is a direct consequence of the arithmetic properties of the limits of a function.
Product theorem limited function to infinitesimal
Product of a function bounded on some punctured neighborhood of point x 0 , to infinitesimal, as x → x 0 , is an infinitesimal function as x → x 0 .
The property of representing a function as the sum of a constant and an infinitesimal function
In order for the function f (x) had final limit, it is necessary and sufficient for
,
where - infinitely small function as x → x 0
.
Properties of infinitely large functions
Theorem on the sum of a bounded function and an infinitely large
The sum or difference of a bounded function on some punctured neighborhood of the point x 0
, and an infinitely large function, as x → x 0
, is infinite great function as x → x 0
.
Theorem on the division of a bounded function by an infinitely large one
If function f (x) is infinitely large as x → x 0
, and the function g (x)- is bounded on some punctured neighborhood of point x 0
, That
.
Theorem on the division of a function bounded below by an infinitesimal one
If a function , on some punctured neighborhood of the point , by absolute value bounded below positive number:
,
and the function is infinitesimal as x → x 0
:
,
and there is a punctured neighborhood of the point on which , then
.
Property of inequalities of infinitely large functions
If the function is infinitely large at:
,
and the functions and , on some punctured neighborhood of the point satisfy the inequality:
,
then the function is also infinitely large at:
.
This property has two special cases.
Let, on some punctured neighborhood of the point , the functions and satisfy the inequality:
.
Then if , then and .
If , then and .
Relationship between infinitely large and infinitesimal functions
From the two previous properties follows the connection between infinitely large and infinitesimal functions.
If a function is infinitely large at , then the function is infinitesimal at .
If a function is infinitesimal for , and , then the function is infinitely large for .
The relationship between an infinitesimal and an infinitely large function can be expressed symbolically:
,
.
If an infinitesimal function has a certain sign at , that is, it is positive (or negative) on some punctured neighborhood of the point , then we can write it like this:
.
In the same way, if an infinitely large function has a certain sign at , then they write:
, or .
Then the symbolic connection between infinitely small and infinitely large functions can be supplemented with the following relations:
,
,
,
.
Additional formulas, linking infinity symbols can be found on the page
"Points at infinity and their properties."
Proof of properties and theorems
Proof of the theorem on the product of a bounded function and an infinitesimal one
Let the function be infinitely large for:
.
And let there be a punctured neighborhood of the point on which
at .
Let us take an arbitrary sequence converging to . Then, starting from some number N, the elements of the sequence will belong to this neighborhood:
at .
Then
at .
According to the definition of the limit of a function according to Heine,
.
Then, by the property of inequalities of infinitely large sequences,
.
Since the sequence is arbitrary, converging to , then, by the definition of the limit of a function according to Heine,
.
The property has been proven.
References:
L.D. Kudryavtsev. Well mathematical analysis. Volume 1. Moscow, 2003.
In order to calculate the sum of a series, you just need to add the elements of the row a given number of times. For example:
In the example above, this was done very simply, since it had to be summed a finite number of times. But what if the upper limit of summation is infinity? For example, if we need to find the sum of the following series:
By analogy with the previous example, we can write this amount like this:
But what to do next?! At this stage it is necessary to introduce the concept partial amount row. So, partial sum of the series(denoted S n) is the sum of the first n terms of the series. Those. in our case:
Then the sum of the original series can be calculated as the limit of the partial sum:
Thus, for calculating the sum of a series, it is necessary to somehow find an expression for the partial sum of the series (S n ). In our particular case, the series is a decreasing geometric progression with a denominator of 1/3. As you know, the sum of the first n elements of a geometric progression is calculated by the formula:
here b 1 is the first element of the geometric progression (in our case it is 1) and q is the denominator of the progression (in our case 1/3). Therefore, the partial sum S n for our series is equal to:
Then the sum of our series (S) according to the definition given above is equal to:
The examples discussed above are quite simple. Usually, calculating the sum of a series is much more difficult and the greatest difficulty lies in finding the partial sum of the series. Featured below online calculator, based on the Wolfram Alpha system, allows you to calculate the sum of fairly complex series. Moreover, if the calculator could not find the sum of the series, it is likely that this series is divergent (in this case the calculator displays a message like "sum diverges"), i.e. This calculator also indirectly helps to get an idea of the convergence of series.
To find the sum of your series, you must specify the series variable, the lower and upper limits summation, as well as the expression for the nth term of the series (i.e., the actual expression for the series itself).