What are the methods to determine the convergence of the series?

First of all, according to the necessary conditions for series convergence, the limit of the general term of series convergence must be zero. Conversely, the limit of the general term is not zero, and the series must not converge.

If the limit of the general term is zero, then continue to observe the characteristics of the general term of the series:

If it is a positive series, you can choose the convergence method of the positive series, such as comparison, ratio, root value, etc.

In the case of a staggered series, it can be based on Leibniz's theorem.

In addition, it can also be judged according to the relationship between absolute convergence and conditional convergence.

1.First, see if the series general term tends to 0. If not, just write "divergent", OK score, and do the next question; If yes, go to 2

2.Look at what series it is, and the staggered series goes to 3; The positive series goes to 4

3.The staggered series uses the Leibniz convergence method, and the decreasing trend of the general term to zero is convergence.

4.The positive series can generally be done by using the ratio convergence method, the comparative convergence method, etc. I can't figure it out to turn 5

5.To see if this series is a definition of an integral, perhaps it can be written in the form of an integral to judge, if the integral is a finite value, it will converge, and vice versa. If you still can't figure it out, turn 6.

6.Write on the volume that "the general term tends to 0, so it can be further discussed". Writing this sentence is a bit of a point. Go back and burn incense to bless passing, over!

The discriminant method for series convergence is as follows:

1. Determine the divergence of the positive series.

1.First, see if the general term of the series tends to zero closure when n tends to infinity (if it is not easy to see, you can skip this step). If it does not tend to zero, the series diverges; If it tends to zero, other methods are considered.

2.Then look at whether the series is a geometric series or a p-series, because the divergence of these two series is known, if it is not a geometric series or a p-series.

3.Ratio discriminant or root value discriminant is used to discriminate.

4.Then use the comparative discriminant method or its limit form to discriminate, and use the comparative discriminant method to discriminate, generally according to the characteristics of the general term to guess its divergence, and then find out the series as a comparison, commonly used as a comparison of the series mainly geometric series and p series.

2. Determine the divergence of the staggered series.

1.The Leibniz discriminant method is used for analysis and judgment.

2.The relationship between the absolute series and the original series is used to make the judgment.

3.In general, if the series diverges, the series may not diverge; However, if the absolute series divergence is determined by the ratio method or the root value method, the series must diverge.

4.Sometimes the series term can be split into two, and the "convergence + divergence = divergence" and "convergence + convergence = convergence" can be used.

3. Find the convergence radius, convergence interval and convergence domain of the power series.

1.If the power of the series is increasing in the order of the natural numbers of x, then its convergence radius is obtained from or , and then the convergence interval of the ground state failure can be written, and then the convergence domain of the power series can be obtained by considering the divergence of the series of several terms at the end of the interval.

2.For the power series of the missing term or the power series of the function of x, the convergence radius can be found according to the ratio discriminant method, or the trembling can be substituted for the power series of t, and then the convergence radius can be obtained.

The methods for judging conditional convergence and absolute convergence are as follows:

A convergent series is said to be absolutely convergent if it still converges after taking the absolute method one by one and the following values; Otherwise, it is said to be conditionally convergent.

A simple comparison series shows that as long as un|Convergence is sufficient to ensure series convergence; Thus the decomposition (not only indicates un|The convergence of implies the convergence of the original series un, and the original series is tabled as the difference between the two convergent positive series.

It can be seen from this that absolute convergence series, like positive series, is much like finite sums, and the order of terms can be arbitrarily changed to sum, and they can be multiplied infinitely distributedly.

Difference Between Conditional Convergence and Absolute Convergence

First, the rearrangement is different.

1. Conditional convergence: The series obtained after any rearrangement of conditional convergence is not conditional convergence, and there are different sums.

2. Absolute convergence: The series obtained after absolute convergence and arbitrary rearrangement is also absolutely convergent, and has the same sum number.

Second, the absolute value is different.

1. Conditional convergence: Conditional convergence diverges from the series (n=1) un after taking the absolute value.

2. Absolute convergence: Absolute convergence takes the absolute value and then converges to the series (n=1) un.

Third, the flaws are different.

1. Conditional convergence: There are flaws in conditional convergence on a,b], so that (b,a)f(x)dx generalized integrals have extreme values.

2. Absolute convergence: Absolute convergence does not have a flaw that would make (b,a)f(x)dx generalized integrals have extremums.

For any term series ( n=1)un, if ( n=1) un converges, then the original series ( n=1)un is said to be absolutely convergent; If the original series (n=1)un converges, but the absolute value diverges to the series (n=1) un, then the original series (n=1)un is said to converge conditionally.

The partial and number series discriminants, the principle of comparison, the comparative discriminant, the radical discriminant, the integral discriminant, the remainder and the Rabe discriminant are used.

For positive series, the comparative discriminant method is a fairly effective discriminant method, by finding a new positive series, comparing the general terms, if the general term of the original series is small, the new series converges, then the original series converges;

If the new series diverges and the original series has a large general term, the original series diverges, and its limit form is usually used in the discriminant process. Limitations: When the series is too complex, it is difficult to determine what the new series is looking for, and the usual approach is to Taylor the general term of the original series to find the equivalent p-series.