求无穷和的极限在高等数学的学习中经常遇到。本文阐述了利用定积分求无穷和极限的原理,并列举了使用定积分求不同类型无穷和极限的方法。Computing Limits of Infinite Sums is often encountered in the study of advanced mathematic...求无穷和的极限在高等数学的学习中经常遇到。本文阐述了利用定积分求无穷和极限的原理,并列举了使用定积分求不同类型无穷和极限的方法。Computing Limits of Infinite Sums is often encountered in the study of advanced mathematics. This article explains the principle of using definite integrals to find infinite sum limits, and provides some methods of solving different types of infinite sum limits by using definite integrals.展开更多
等价无穷小是极限理论的一个重要组成部分,选取合适的等价无穷小代换,可以极大地简化极限问题的处理。使用等价无穷小需要满足一定的条件,很多学习者对等价无穷小代换的使用条件认识不深,经常错用等价无穷小代换。针对这个问题,本文通...等价无穷小是极限理论的一个重要组成部分,选取合适的等价无穷小代换,可以极大地简化极限问题的处理。使用等价无穷小需要满足一定的条件,很多学习者对等价无穷小代换的使用条件认识不深,经常错用等价无穷小代换。针对这个问题,本文通过等价无穷小的本质对等价无穷小代换的使用条件进行解析,使学习者能够充分认识并理解等价无穷小的使用条件,理解和掌握等价无穷小的应用,对于深入学习和应用微积分知识具有重要的作用。Equivalent infinitesimal is an important component of limit theory, and selecting appropriate equivalent infinitesimal substitutions can greatly simplify the handling of limit problems. The use of equivalent infinitesimal substitution requires certain conditions to be met, and many learners have a limited understanding of the conditions for using equivalent infinitesimal substitution and often misuse it. In response to this issue, this article analyzes the usage conditions of equivalent infinitesimal substitution through the essence of equivalent infinitesimal, enabling learners to fully understand and comprehend the usage conditions of equivalent infinitesimal, understand and master the applications of equivalent infinitesimal, which plays an important role in in-depth learning and application of micro integration knowledge.展开更多
定义在半无穷区间Z=[ 14,+∞ )上的有理函数系({ g2n:n∈N },其中N表示非负整数全体),给出了区间Z上的希尔伯特空间(L2(Z,w))是研究者最新提出的研究内容。该空间中的基定理证明了{ g2n:n∈N }是L2(Z,w)空间中的一组基。其定理的证明采...定义在半无穷区间Z=[ 14,+∞ )上的有理函数系({ g2n:n∈N },其中N表示非负整数全体),给出了区间Z上的希尔伯特空间(L2(Z,w))是研究者最新提出的研究内容。该空间中的基定理证明了{ g2n:n∈N }是L2(Z,w)空间中的一组基。其定理的证明采用了等距同构的方法,未详细展开。本文首先给出L2(Z,w)空间中的另一个刻画,并利用此刻画,采用函数论的方法,给出基定理的一个新证明,以展现此空间中的函数逼近结构。A family of rational functions { g2n:n∈N }(where N denotes the set of all nonnegative integers) defined on a semi-infinite interval Z=[ 14,+∞ ), and a Hilbert space (L2(Z,w)) on Zis a recently proposed research subject. { g2n:n∈N }was proved to be an orthonormal basis for L2(Z,w)in a theorem (the basis theorem) by using an isometry. The original proof is so brief that it might not have shown the hierarchy of function approximation relations clear enough. In this paper we give another characterization and raise examples of functions of several kinds in it. By taking advantages of this characterization, and by applying a function theory method, we offer a new proof for the basis theorem. We wish our deduction could show the hierarchical structurer of the function approximation in this space.展开更多
基金Supported by the National Natural Science Foundation of China(12171335,12301603)the Science Development Project of Sichuan University(2020SCUNL201)the Scientific Foundation of Nanjing University of Posts and Telecommunications(NY221026)。
文摘求无穷和的极限在高等数学的学习中经常遇到。本文阐述了利用定积分求无穷和极限的原理,并列举了使用定积分求不同类型无穷和极限的方法。Computing Limits of Infinite Sums is often encountered in the study of advanced mathematics. This article explains the principle of using definite integrals to find infinite sum limits, and provides some methods of solving different types of infinite sum limits by using definite integrals.
文摘等价无穷小是极限理论的一个重要组成部分,选取合适的等价无穷小代换,可以极大地简化极限问题的处理。使用等价无穷小需要满足一定的条件,很多学习者对等价无穷小代换的使用条件认识不深,经常错用等价无穷小代换。针对这个问题,本文通过等价无穷小的本质对等价无穷小代换的使用条件进行解析,使学习者能够充分认识并理解等价无穷小的使用条件,理解和掌握等价无穷小的应用,对于深入学习和应用微积分知识具有重要的作用。Equivalent infinitesimal is an important component of limit theory, and selecting appropriate equivalent infinitesimal substitutions can greatly simplify the handling of limit problems. The use of equivalent infinitesimal substitution requires certain conditions to be met, and many learners have a limited understanding of the conditions for using equivalent infinitesimal substitution and often misuse it. In response to this issue, this article analyzes the usage conditions of equivalent infinitesimal substitution through the essence of equivalent infinitesimal, enabling learners to fully understand and comprehend the usage conditions of equivalent infinitesimal, understand and master the applications of equivalent infinitesimal, which plays an important role in in-depth learning and application of micro integration knowledge.
文摘定义在半无穷区间Z=[ 14,+∞ )上的有理函数系({ g2n:n∈N },其中N表示非负整数全体),给出了区间Z上的希尔伯特空间(L2(Z,w))是研究者最新提出的研究内容。该空间中的基定理证明了{ g2n:n∈N }是L2(Z,w)空间中的一组基。其定理的证明采用了等距同构的方法,未详细展开。本文首先给出L2(Z,w)空间中的另一个刻画,并利用此刻画,采用函数论的方法,给出基定理的一个新证明,以展现此空间中的函数逼近结构。A family of rational functions { g2n:n∈N }(where N denotes the set of all nonnegative integers) defined on a semi-infinite interval Z=[ 14,+∞ ), and a Hilbert space (L2(Z,w)) on Zis a recently proposed research subject. { g2n:n∈N }was proved to be an orthonormal basis for L2(Z,w)in a theorem (the basis theorem) by using an isometry. The original proof is so brief that it might not have shown the hierarchy of function approximation relations clear enough. In this paper we give another characterization and raise examples of functions of several kinds in it. By taking advantages of this characterization, and by applying a function theory method, we offer a new proof for the basis theorem. We wish our deduction could show the hierarchical structurer of the function approximation in this space.
