1. Describe the regularity: A series is a set of numbers arranged according to certain laws, which can be used to describe and study various phenomena and laws. For example, arithmetic progression can describe the position change of an object in uniform motion; Geometric series can describe the process of compound interest growth.
2. Mathematical modeling: Sequence is one of the basic tools for mathematical modeling. By establishing series model, practical problems can be transformed into mathematical problems for theoretical analysis and solution. For example, the series model can be used to study population growth, resource consumption and other issues.
3. Mathematical analysis: Sequence is one of the important contents of mathematical analysis. Through the study of sequence, we can deeply understand the mathematical concepts and methods such as limit and calculus. For example, infinitesimal and infinity can be defined by using the limit of sequence, thus introducing the concept of calculus.
4. Functional relationship: There is a close relationship between sequence and function. A sequence can be regarded as a function in discrete cases, while a function can be regarded as a sequence in continuous cases. Through the study of sequence and function, we can reveal the connection and difference between them, and further enrich the mathematical theory.
5. Algorithm design: Sequences also have important applications in computer science and algorithm design. For example, Fibonacci series can be used to design recursive algorithms; Yang Hui triangle can be used to solve the combination problem.
In a word, the application of sequence in mathematical research is very extensive and important, and it is the basis and key to understand and study various mathematical problems. Through the study of sequence, we can deeply understand the essence and laws of mathematics and provide powerful mathematical tools and methods for solving practical problems.