What is the difference between a convergence domain and a convergence interval

There is only one difference between the convergence domain and the convergence interval: whether the interval is closed or not.

The convergence interval is an open interval, and the convergence domain is to determine whether there is convergence at the end of the convergence interval. For example, if the convergence radius of a series is found to be 5, then the convergence interval is (-5,5), and the next step is to find the convergence domain with x -5 and x 5, respectively, to see if they converge.

If the convergence radius of the power series is r, then the convergence interval (-r,r) is directly concluded regardless of the endpoint convergence. If further discussion is about the convergence of the series at the point -r or r, such as convergence at the point -r and non-convergence at the point r, then the convergence domain of the power series is called [-r,r].

For example, if both converge at points -r and r, then the convergence domain of the power series is called [-r,r], and if the points -r,r do not converge, then the convergence domain of the power series is still (-r,r).

In short, the convergence interval is obtained directly from the convergence radius, and the convergence domain is the conclusion after discussing the convergence at both ends of the convergence interval. The convergence interval may be the same as the convergence domain and may be a subset of the convergence domain.

First, the concept is different.

The convergence domain is a concept in the chapter on function series, which represents the set of all convergence points of a function series, which refers to the convergence at one point and the approach to a certain value. The convergence types are convergent sequence, function convergence, global convergence, and local convergence.

The convergence interval is a concept in the power series chapter, which is the open interval (-r, r), where r is the convergence radius.

Second, the opening and closing of the interval is different.

Convergence domain: It can be an open interval or a closed interval. To determine the absolute convergence radius of the series, the convergence at the endpoints, and whether the endpoints are desirable, it may be an open interval, a closed interval, or a semi-open and semi-closed interval to determine the convergence domain.

Convergence Range: Open Range. It is expressed as the open interval of (-r,r) without discussing the convergence radius and the situation at the endpoint.

Third, the judgment of the conclusion is different.

The convergence interval is obtained directly from the convergence radius, and the convergence domain is the conclusion after discussing the convergence at both ends of the convergence interval. The convergence interval may be the same as the convergence domain and may be a subset of the convergence domain.

Do you still have to take the convergence test? The convergence interval is to consider whether the endpoint of the interval is desirable or not, it may be an open interval, it may be a closed interval, or a semi-open and semi-closed interval, while the convergence interval is an open interval of (-r,r).

The convergence of the series of function terms can be divided into: consistent convergence, and point-by-point convergence, and the convergence mentioned by the engineering graduate school is the property of point-by-point convergence.

The convergence domain is the set that guarantees its point-by-point convergence, for example, on [0,1), which means that the endpoint can also be a convergence point.

The convergence interval is the interval that guarantees the continuity of the sum function, because the general term is continuous, and the sum function is continuous, so it may be that u n(x) is continuous on [1,1], but the sum function n (x) is continuous on (1,1), and the problem of continuity at the endpoint depends on the nature of consistent convergence.

Difference Between Convergence Function and Bounded Function:

First, definitions. The convergence function refers to the domain defined by the fetch function.

Any number in the function is brought in, and the result is convergent. The cut-off value of the convergence function in the defined domain is actually the maximum and minimum value of the function in the defined domain. A bounded function means that the values in the function definition field are brought into the function relation, and the results obtained change in the same interval, which we call the function bounded.

Keep an eye on the suspicion of defeat. Second, the relationship between the two, the convergence function is contained in the bounded function, that is, the range of the bounded function is larger, and the range of the convergence function is smaller than that of the bounded function, if a function has been determined to be a bounded function, then there are two cases of whether it is a convergence function or not, it may be a convergence function, it may not be a convergence function, but if a function is a convergence function, then it must be a bounded function. The relationship between the two is that of inclusion and inclusion.

Third, the judgment of the conclusion is different, and the convergence interval is directly based on the convergence radius.

The convergence domain is the conclusion after discussing the convergence at both ends of the convergence interval. The convergence interval may be the same as the convergence domain and may be a subset of the convergence domain.

Function convergence. It is defined in a similar way to series convergence. Cauchy Convergence Criterion.

About the definition of the convergence of the function f(x) at the point x0. For any real number b>0, there is c>0, and for any x1, x2 satisfies 0<|x1-x0|Mathematical analysis.

The essence of the spirit.

If given a function column defined on interval i, u1(x), u2(x) , u3(x)...to un(x).is an expression made up of this function column.

u1（x）+u2（x）+u3（x）+.un（x）+.It is called an infinite series (function term) defined on interval i.

Abbreviated (function item) series.

Find the convergence radius r, written as an open interval in the form of (-1,1), this is called the convergence interval Lingjin.

The interval breakpoint is brought into the scale commetic base to judge the divergence.

If x=-1 converges and x=1 diverges, then the convergence domain is the finger hole x -1,1).

Hello! If the convergence radius is zero, there is only one point in the convergence domain, which is 0 or x0. The Economic Mathematics team will help you with the answers, please be in time. Thank you!

General derivation.

Divide the n+1 term by the nth term, the absolute value of the whole is less than 1, and the absolute value of x (or x-a, which depends on your series) is less than the convergence radius.

The convergence domain is the area where all the points that converge are found.

For example, if the radius of convergence is r, finding the convergence domain is to determine the logarithm of x (or x-a).

1. The interval closure is different

The convergence interval is an open range.

The convergence domain is to determine whether there is convergence at the end of the convergence interval.

If the radius of convergence of the power series.

is r, then the convergence interval (-r,r) is directly concluded regardless of the endpoint convergence. If further discussed, the convergence of the series at the point -r or r.

Second, the convergence is different:

In the convergence domain, we must pay attention to the convergence of the endpoint, determine whether the endpoint is convergent, and then determine the opening and closing of this interval. If this endpoint is convergent, then be sure to include this point when writing the convergence domain, i.e., close it at this endpoint.

Therefore, the convergence domain may be an open interval (i.e., both endpoints are divergent), a semi-closed and semi-open interval (i.e., converge at the closed junction point), or a fully closed interval (i.e., both endpoints are pure and convergent).

Question 1: What is a convergence domain? What is a Discrete Sequence? A convergence domain is a sequence that converges within the range of this domain, and a discrete sequence means that when n approaches n or infinity, the extreme value of the sequence is greater than 1

Question 2: What is the convergence domain used for? What is the convergence domain used for?

Well, first of all, there is no convergence domain for constant term series, they either converge or diverge. The function term series, which contains an unknown x, is equivalent to a function, and the series converges only if the function x is in the convergence domain, and diverges when it is not in the convergence domain (two endpoints to consider).

If you really ask what the convergence domain is for, it is the effect of the range of x values in order to converge the series.

Question 3: What is the difference between the convergence domain and the convergence interval of the power series, and how to find the convergence domain and the convergence interval respectively The convergence interval is an open interval, and the convergence domain is to determine whether it converges at the end of the convergence interval.

For example, if you find a series with a convergence radius of 5, then the convergence interval is not open (-5, 5), and the next step is to find the convergence domain with x -5 and x 5, respectively, to see if it converges.

For example, if x -5 converges and x 5 diverges, then the convergence domain is [-5,5).

Question 4: What is the convergence domain The convergence domain is a sequence that converges within the scope of this domain.

Question 5: How to find the convergence domain of the function, what are the steps 20 points The easiest way is to intuitively look at the type of the function, judge the image of the function, and judge the convergence divergence according to the image.

Question 6: What is the specific difference between the convergence domain and the convergence interval of a power series? Assuming that the convergence radius r of the power series has been found, the convergence interval of the power series is the open interval (-r,r);

Then to determine whether the power series converges at x= -r and x=r, and take these two points, that is, the two endpoints of the open interval (-r, r), into account, which is the convergence domain.

For example, if it converges at x= -r and diverges at x=r, the convergence domain is [-r,r].

Question 7: What is the difference between the convergence domain and the convergence interval The convergence domain considers the end virtual check answer point, and the convergence interval does not consider.

Divide the n+1 term by the nth term, and the absolute value of the whole is less than 1, and the absolute value of x (or x-a, which depends on your series) is less than the convergence radius. The convergence domain is the area where all the points that converge are found.

1. The radius of convergence r is a non-negative real number or infinity such that in |z -a|r time the power series diverges. Power series, one of the important concepts in mathematical analysis, means that each term of the series is a constant multiple (x-a) of the nth power corresponding to the series term n (n is an integer counted from 0, and a is a constant).

2. If the power in the power series is increasing in the order of natural numbers, that is, the series is a power series that is not missing, two methods, namely the coefficient modulus ratio value method and the coefficient modulus root value method, can be used to find its scale convergence radius r. If the powers in a power series are not increasing sequentially in the order of natural numbers (for example, unodd powers, or even powers, etc.), the ratio convergence method must be used directly.

3. Because the convergence domain of the function term series is actually composed of all the convergence points, and for the determination of the convergence of the function term series corresponding to each convergence point, it actually corresponds to the determination of the convergence of the constant series, so the calculation of the convergence domain of the function term series is generally based on the method of constant series judgment, and the commonly used are the ratio convergence method and the root value discrimination method based on the absolute value of the term.