Cauchy inequality formula and inference are as follows:

1. Cauchy inequality formula: For any real number sequence (a_i) and (b_i), there is (∑ A _ I 2) * (∑ B _ I 2) ≥ (∑ A _ I * B _ I).

2. Cauchy inequality inference: For any nonnegative real number sequence (a_i) and (b_i), there are (A _ 1 2+A _ 2 2+...+A _ n 2) * (B _ 1 2+B _ 2 2+...

3. for any nonnegative real number sequence (a_i) and (b_i), there are (a _ 1 2+a _ 2 2+...+a _ n 2) * (b _ 1 2+b _ 2 2+...+b _ Where m is a positive integer.

The value of Cauchy inequality formula

1. Generalization of basic inequality: Cauchy inequality can be regarded as the generalization of basic inequality. It allows us to estimate and prove the inequality of any real number sequence in a wider situation. This enables us to have more tools and skills when dealing with more complicated mathematical problems.

2. Solving optimization problems: Cauchy inequality can be used to solve some optimization problems. For example, under some constraints, how to choose a real number sequence to make a function reach the minimum or maximum can be solved by Cauchy inequality. This method is widely used in operational research, cybernetics and other fields.

3. communication discreteness and continuity: Cauchy inequality is expressed in both discrete and continuous forms. This enables us to build a bridge between discrete and continuous fields and better understand and communicate the relationship between these two fields.

4. Combination of theoretical analysis and practical application: The theoretical analysis of Cauchy inequality is closely combined with practical application. For example, in information theory, Cauchy inequality is used to prove the reciprocal relationship between quantity and quality of information; In probability theory, Cauchy inequality is used to estimate the expectation and variance of random variables; In physics, Cauchy inequality is used to describe the uncertain relationship in quantum mechanics.