Cauchy convergence principle is one of the important theorems in mathematical analysis, which provides a new idea and method for studying the limit of sequence and function.
With the definition of limit, in order to judge whether a particular sequence or function has limit, people must constantly discuss the necessary and sufficient conditions for the existence of limit. Through the continuous efforts of many mathematicians, French mathematician Cauchy finally got a perfect result. We will describe it in the form of a theorem called Cauchy convergence principle.
Theorem statement:
The necessary and sufficient conditions for the sequence {xn} to have a limit are: for any given ε >; 0, with a positive integer n, when m, n >；; When n, there is | xn-XM | < ε holds.
When Cauchy convergence principle is extended to function limit, there are:
The necessary and sufficient conditions for the function f(x) to have a limit at infinity are: for any given ε >; 0, where z belongs to a real number, when x, y >; When z, | f (x)-f (y) | < ε holds.
In addition, Cauchy convergence principle can also be extended to judge whether generalized integrals converge or not and whether series terms converge, which has a wide range of applications.
Examples of evidence:
It is proved that xn =1-1/2+1/3-1/4+...+[(-1)]/n has a limit.
Prove that for any m, n belongs to a positive integer, m >；; n
| xn-XM | = |[(- 1)^(n+2)]/(n+ 1)+......+[(- 1)^(m+ 1)]/m |
When m-n is odd | xn-XM | =| [(-1) (n+2)]/(n+1)+...+[(-1)]]
& lt 1/n(n+ 1)+ 1/(n+ 1)(n+2)+......+ 1/(m- 1)m
=( 1/n- 1/m)→0
{xn} Convergence of Cauchy Convergence Principle
When m-n is even | xn-XM | =| [(-1) (n+2)]/(n+1)+...+[(-1)]]
& lt 1/n(n+ 1)+ 1/(n+ 1)(n+2)+......+ 1/(m-2)(m- 1)- 1/m
=( 1/n- 1/(m- 1)- 1/m)→0
{xn} Convergence of Cauchy Convergence Principle
To sum up, {xn} converges, that is, {xn} has a limit.
Is it comprehensive enough?