高等数学中无穷小量的相关概念解析及应用

Analysis and Application of the Related Concepts of Infinitesimal in Advanced Mathematics

通过对无穷小认识的曲折历程的叙述揭示了无穷小蕴含的哲学思辨色彩,从无穷小阶数比较的角度阐释了一元函数导数与微分的本质与联系,利用泰勒公式比较无穷小的阶数并解释等价无穷小的实质,指出等价无穷小替换的使用错误根源在于用于替换的泰勒多项式精度达不到所需要求,最后给出了无穷小比较在极限审敛法中的应用。

Through the description of the tortuous process of understanding infinitesimals,the philosophical speculative color contained in infinitesimals is revealed and its connotation is explained.From the perspective of comparing the order of infinitesimals,the essence and relationship between the derivative and differential ofunary function are explained.The Taylor formula is used to compare the order of infinitesimals and explain the essence of equivalent infinitesimals.It is pointed out that the root cause of the error in the use of equivalent infinitesimals is that the accuracy of the Taylor polynomial used for substitution cannot meet the required requirements.Finally,the application of infinitesimal comparison in the limit convergence method is presented.