In the end, how to understand the concept of the limit of the number series urgently

Establish. xn}

is a series of real numbers, a

is a definite number If there is a positive number given to any given number, there is always a positive integer n, such that dang.

There is an XN-A < when n>n

is called a sequence. converges at a, a definite number.

A is called a sequence.

xn}.

In fact, it means that the sequence tends to be a number, and this number is the limit of the sequence.

n>n means that not every term in the sequence tends to this number, but all terms that must follow one term in the sequence tend to this number.

For example, the series, -1, 3, 4, -3, -5, 6, 1 2, 1 3, 1 4, 1 5, etcThere is no rule in the terms at the beginning of this series, but starting from the term 1 2, the following terms tend to 0, and the limit of all the terms in this series is 0, that is, n>6, at this time, n=6, satisfying xn-a <

The definition of the limit is that no matter how small an integer m is, there exists a number n1, and when n > n1, |xn-a|Questions 1 and 2 should both be |xn-a|Smaller and smaller.

The limits of judgment should be strictly defined by definition.

In the example, xn-a is getting closer and closer to 1, which does not explain the problem, but I personally think that it is like this: the series of numbers has a tendency to have the previous finite terms that do not tend to the limit value, so it cannot be said to be n

When it is bigger and bigger, it is getting smaller and smaller, and it should be said that n is greater than a certain value, because the previous terms are indefinite, at this time, when n

As it grows larger, the changes are uncertain. To be clear: they also have no effect on limit values. I think that's what the definition of limits is about.

The third floor is also not right, the xn-a in the example you gave is getting closer and closer to 1, which does not explain the problem, I personally think it is like this: the series of numbers has a tendency to have the previous finite terms and do not tend to the limit value, so it cannot be said that it is n

When it is bigger and bigger, it is getting smaller and smaller, and it should be said that n is greater than a certain value, because the previous terms are indefinite, at this time, when n

As it grows larger, the changes are uncertain. To be clear: they also have no effect on limit values. I think that's what the definition of limits is about. Who has a better answer, communicate.

Hello! If you want to be wrong, just give a counter-example.

Let xn=1+1 n,a=0.

So xn-a is getting smaller and smaller, and xn-a is getting closer and closer to 0, but obviously a is not the limit of xn.

Let the number series be when n

Bigger and bigger, smaller and smaller, then.

limxn=a

n is clearly false, e.g. xn=-n

Well, n

When you say -n-a is not getting smaller and smaller!!

Let the sequence of numbers, when n gets bigger and bigger, and xn-a gets closer and closer to 0, then limxn=an is obviously wrong as well.

For example: xn=2+(1 n).

You yourself say that 2+(1 n)-1 is not getting closer to 0

Definition of the limit of the sequence:Logarithmic sequence, if there is a constant a, for any.

0, there is always a positive integer.

n, such that when n > n, |xn-a|< True, then it is clear that a is the limit of the number series.

Proof that for any c >0, the inequality is solved.

1/ vn|=1/ vn<ε

Get n>1 2, take n=[1 2]+1.

Thus, for any >0, there is always a natural number.

Take n=[1 2]+1.

When n > n, there is | 1/n| <

Therefore, 1im(n-> 1 j n)=0.

Conditions for the existence of the limit of the sequence: monotonic defined theoremIn the real number system, the bounded monotonic bounded sequence must have a limit. The compactness theorem that any bounded sequence must have convergent subcolumns.

Applications of the Sequence Limit:

Let it be a converging series.

And: When n tends to infinity, the limit of the sequence, is aIf n exists such that when n > n, there are xn yn zn, then the series converges and the limit is a

It is suitable for solving the limits of functions that cannot be directly solved or trapped by the limit algorithm.

Indirectly, the limit of f(x) is determined by finding the limits of f(x) and g(x).

Definition of grip wheel for the limit of the sequence:

The number series has a limit, that is, when n tends to infinity, the term xn of the series is infinitely close to or equal to a, and taking an arbitrary value is to show that no matter how small the number is, the difference between xn and a is always less than , that is, xn infinitely approaches or is equal to a.

When looking at n>n, note that the original words are: ......For arbitrarily small , there is always a positive integer n such that when n > n, |xn-a|< This shows that no matter how small, when n is large enough, it can be satisfiedxn-a|<ε

That is, even if it is very small (approaching 0), when n is large enough (tends to infinity) it will satisfy that the difference between xn and a is less than (approaching 0).

Expansion: Conditions for the existence of limits:

Monotonic Defined Theories In the real number system, there must be a limit to the monotonous bounded number sequence.

Compactness Theorem Any bounded sequence must have converging subcolumns.

The idea of limit is an important idea of modern mathematics, and mathematical analysis is a discipline that studies functions based on the concept of limit and the theory of limit (including series) as the main tool. The so-called limit idea refers to "a mathematical idea that uses the concept of limit to analyze and solve problems".

The concept of the limit of a sequence is that if a sequence of numbers tends infinitely towards a certain real number, then the definite real number is called the limit of the sequence.

A sequence is a sequence of ordered numbers with a set of positive integers (or a finite subset of it) as the defined domain. Each number in the sequence is called an item in the sequence. The number in the first place is called the first term of the series (usually also called the first term), the number in the second place is called the second term of the series, and so on, the number in the nth position is called the nth term of the series, which is usually denoted by an.

Famous sequences include Fibonacci sequences, Cattelan numbers, Yang Hui triangles, etc.

"Equal sum sequence" means that in a sequence, if the sum of each term and its next term is the same constant, then the sequence is called the equal sum sequence, and this constant is called the common sum of the sequence. For a series of numbers, if the sum of any of its consecutive k terms is equal, we call the series an equal sum sequence, and its nature is that it must be a circular sequence.

Proportional sequences are also often used in life.

Limit connotation:

"Limit" is the basic concept of calculus, a branch of mathematics, and "limit" in a broad sense means "infinitely close and never reachable". The "limit" in mathematics refers to a certain variable in a function, which gradually approaches a certain definite value a in the process of changing to a certain value in the process of becoming larger or smaller, and "can never coincide to a", "can never be equal to a, but takes equal to a, which is enough to obtain high-precision calculation results".

The change in this variable is artificially defined as "always approaching without stopping", and it has a "tendency to constantly get extremely close to point A". Limit is a description of a "state of change". The value a that this variable is always approaching is called the "limit value" and can also be represented by other symbols.

The idea of limit is an important idea of modern mathematics, and mathematical analysis is a discipline that studies functions based on the concept of limit and the theory of limit (including series) as the main tool.

The so-called limit idea refers to "a mathematical idea that uses the concept of limit to analyze and solve problems". For the unknown quantity to be examined, first try to correctly conceive another variable related to its change, and confirm the influence of this variable through the process of infinite change, and the trend result is a very precise approximate equal to the unknown quantity sought; Using the limit principle, the results of the unknown quantities under investigation can be calculated.

The idea of limits is the basic idea of calculus, which is a series of important concepts in mathematical analysis, such as the continuity of functions, derivatives (0 to get the maximum or minimum), and definite integrals, which are defined with the help of limits.

Let it be a sequence, and if the limb is in constant a, there will always be a positive integer for any given positive number, no matter how small.

n, such that when n > n, the inequality epoch is simplified.

If it is true, then the constant a is the limit of the sequence, or the sequence converges at a, which is recorded as.

A geometric explanation

From Tongji University.

Let be a series of real numbers, a is a definite number If there is always a positive integer n for any given positive number, so that when n>n there is xn-a n means that this number is not necessarily every term in the cover column tends to this number, but Sun Hao must have all the terms after a certain term in the number tend to this number.

For example, the series, -1, 3, 4, -3, -5, 6, 1 2, 1 3, 1 4, 1 5, etcThere is no rule of chaos at the beginning of this series, but starting from the term 1 2, the following terms tend to 0, and the limit of all the terms in this series is 0, that is, n>6, at this time, n=6, satisfying xn-a

Limits can be divided into sequence limits and function limits, and the first step in learning calculus is to understand the necessity of introducing "limits": because algebra is a familiar concept, but algebra cannot deal with the concept of "infinity".

Therefore, in order to use algebraic processing to represent infinite quantities, the concept of "limit" was carefully constructed. In the definition of "limit", we can see that this concept bypasses the trouble of dividing a number by 0 and introduces a hail process of any small amount.

Apply. In daily life, people often use the difference series, such as: when classifying the size of various products, when the maximum size is not much different from the minimum size, it is often graded according to the difference series.

If it is an equal difference series and there is an=m and am=n, then am+n=0. Ji Jinsheng.

Its application in mathematics can be used for example: quickly calculate how many integer multiples of 6 between 23 and 132, there is more than one algorithm, here is the introduction of using the number series to calculate the first term of the equal difference series a1 = 24 (24 is 4 times of 6), equal difference d = 6;So let an=24+6(n-1)<=132 solve n=19.

How to understand the definition of the limit of the common number series? Candidates who are learning this knowledge point of Xunhe can take a look, and I have prepared for you below "How to understand the definition of the limit of the number series", for reference only, I wish you a happy reading!

How to understand the definition of the limit of a sequenceThe mu envy limit is that when n increases infinitely, an infinitely approaches a certain constant;

That is, when n is large enough, |an-a|can be arbitrarily small, less than the positive number e I give;

That is, when n is greater than a positive integer n, |an-a|can be less than the given positive number e;

That is, for any e>0, there is a positive integer n, and when n > n, |an-a|。

Further reading: Definition and properties of the limits of the sequenceSequence limit definitionDefinition: Set |xn|For a sequence, if there is a constant a for any given positive number (no matter how small), there is always a positive integer n, such that when n > n, the inequality |xn - a|< is true, then the constant a is a sequence|xn|The limit, or sequence|xn|converges on a.

Denoted as lim xn = a or xn a(n).

The nature of the limits of the sequence1.Uniqueness: If the limit of the sequence exists, then the limit value is unique;

2.Changing the finite term of a series does not change the limit of the series.

The limits of several commonly used sequences:

an=c constant series The limit is c;

an=1 n The limit is 0;

an=x n absolute value x is less than 1 and the limit is 0.

Definition of the limit standard of the number series: for the logarithmic sequence, if there is a constant a, and the search town for any >0, there is always a positive integer n, such that when n > n, |xn-a|< is true, then a is said to be the limit of the sequence.

Proof of Solution: For any >0, the inequality is solved.

1/√n│=1/√n<ε

Get n>1 take n=[1 1.

Therefore, for any >0, there is always a natural number n=[1 1.

When n > n, there is 1 trace of the missing brother n <

Therefore, lim(n-> 1 n)=0.