One,

1. Divergence and convergence are just a limit concept for sequences and functions. Generally speaking, when variables tend to infinity, if the values of their general terms tend to a certain value, the sequence or function is convergent, and it can be judged whether it is convergent only by asking their limits. To prove whether a sequence converges or diverges, just use the theorem in the book.

2. For series, it is also a concept of limit, but the difference is that this limit is for the partial sum of series, and it is only necessary to judge whether a series converges according to the judgment method in the book.

Second,

The order of 1. convergence sequence is a series, and a is a fixed real number, if for any given B >；; 0, with a positive integer n, so that for any n >；; N, you have | an-a | < B, the sequence has limit a, and the sequence is called convergence. Non-convergent sequences are called divergent sequences.

2. The definition of convergence function is similar to the convergence of sequence. Cauchy Convergence Criterion: On the Convergence Definition of Function f(x) at Point x0. For any real number b>0, c > exists; 0, for any x 1, x2 satisfies 0.

Convergence sequence

Let {} be a series, and A be a fixed real number, if for any given b >；; 0, with a positive integer n, so that for any n >；; N, you have |-a | < B is a constant, that is, the sequence {} converges to a (the limit is a), that is, the sequence {} is convergent sequences.

Function convergence

Defined in a way similar to sequence convergence. Cauchy Convergence Criterion: On the Convergence Definition of Function f(x) at Point x0. For any real number b>0, c > exists; 0, for any x 1, x2 satisfies 0.

The definition of convergence well embodies the spiritual essence of mathematical analysis.

Given a function sequence defined on the interval I, u 1(x), u2(x), u3(x) ... to the United Nations (x) ... and then the expression U 1 (x)+U2 (x)+U3 (x)+

Remember that rn(x) = s (x)-sn (x), and rn(x) is called the remainder of the function series term (of course, only x is meaningful in the convergence domain, lim n→∞rn (x)=0.

Convergence and divergence of iterative algorithm

1. Global convergence

For any X0∈[a, b], the sequence of points generated by the iterative formula Xk+ 1=φ(Xk) converges, that is, when k→∞, the limit of Xk tends to X*, then Xk+ 1=φ(Xk) is called in [a, b].

2. Local convergence

If X* exists in a neighborhood and r = {x || x-x *|| < δ}}, the point sequence generated by Xk+ 1=φ(Xk) converges for any X0∈R, so Xk+ 1=φ(Xk) converges. ..

In mathematical analysis, the concept opposite to convergence is divergence. Divergent Series refers to non-convergent series (in Cauchy sense). For example, the sum of series, that is, the part and sequence of any series have no finite limit.

If a series is convergent, the terms of this series will definitely tend to zero. Therefore, any series whose term does not tend to zero is divergent. However, convergence is a stronger requirement than this: not every series whose term tends to zero converges. One of the counterexamples is that the divergence of harmonic series was proved by medieval mathematician oris.

References:

Baidu Encyclopedia-Fusion? Baidu encyclopedia-divergence