摘要泰勒展开式在高等数学分析学中占据着非常重要的重要内容，本文从三个大

Abstract <br>Taylor expansion plays a very important role in advanced mathematical analysis. This article discusses Taylor expansion in three aspects. First, the Taylor expansion is proved by the method of undetermined coefficients. It is understood that the Taylor expansion is derived using the principle of the incremental method, and the two residual terms of the Taylor expansion, namely the Piano type residual and the pull, are discussed. The remainder of the Grignard type thus derives the application of Taylor expansion in mathematical analysis. In the discussion of the application of Taylor expansion in mathematical analysis, this article mainly introduces the Taylor formula in mathematical analysis from seven aspects: seeking limits, proving inequalities, determining the convergence of series and generalized integrals, and performing approximate calculations. In some applications. Then, the content of the mathematical analysis of Taylor expansion is combined with knowledge of higher algebra. By discussing the determinant representation of Taylor expansion and the method of solving the determinant by Taylor expansion, the Taylor expansion The application in the field of higher algebra cleverly combines the knowledge of analysis with algebra to provide new ideas for solving some problems of higher algebra. The importance of mathematics learning. <br>Keywords: Taylor expansion, remainder, mathematical analysis, advanced algebra, application

摘要<br>泰勒展开式在高等数学分析学中占据着非常重要的重要内容，本文从三个大的方面对泰勒展开式进行了比较深入的讨论。首先，用待定系数的方法证明了泰勒展开式，了解到泰勒展开式是利用增量法原理进行推导而来的，以及讨论了泰勒展开式的两种余项即皮亚诺型余项和拉格朗日型余项，因而得出了泰勒展开式在数学分析中的应用。在泰勒展开式在数学分析里的应用的问题讨论中，本文主要从求极限，证明不等式，判定级数和广义积分的敛散性以及进行近似计算等一共七个方面介绍了泰勒公式在数学分析中的一些应用。紧接着，又将泰勒展开式这一数学分析中的内容同高等代数的知识相结合，通过讨论泰勒展开式的行列式表示法以及用泰勒展开式求解行列式的方法，介绍了泰勒展开式在高等代数领域内的应用，巧妙地将分析学的知识同代数学结合起来，提供了解决部分高等代数问题的新思路，使我们更加清楚的认识到泰勒展开式在计算中的强大作用，以及在数学学习中的重要意义。<br>关键词：泰勒展开式 证明 余项 数学分析 高等代数 应用

abstract<br>Taylor expansion plays an important role in higher mathematics analysis. In this paper, Taylor expansion is discussed from three aspects. First of all, the Taylor expansion is proved by the method of undetermined coefficient. It is known that the Taylor expansion is derived by the principle of incremental method. Two kinds of remainder of the Taylor expansion are discussed, i.e. piano type and Lagrange type. Therefore, the application of the Taylor expansion in mathematical analysis is obtained. In the discussion of the application of Taylor's expansion in mathematical analysis, this paper mainly introduces the application of Taylor's formula in mathematical analysis from seven aspects, such as finding the limit, proving the inequality, judging the convergence and divergence of series and generalized integral, and carrying out approximate calculation. Then, combining the contents of the mathematical analysis of Taylor expansion with the knowledge of higher algebra, this paper introduces the application of Taylor expansion in the field of higher algebra by discussing the expression of the determinant of Taylor expansion and the method of solving the determinant with Taylor expansion, ingeniously combines the knowledge of analysis with algebra, and provides some solutions to higher algebra The new thinking of the problem makes us more clearly realize the powerful role of Taylor expansion in calculation and the important significance in mathematics learning.<br>Key words: Taylor expansion to prove the application of Higher Algebra in remainder mathematical analysis