序列与级数

学分：3

本模块课程通过阐释离散情况下的收敛概念来补充“实分析”核心课程的内容。以实数序列为基本函数，本课程深入探讨了收敛级数的基本概念。其重点是数学证明以及基础技能和技巧的培养，这些都是当代数学的重要内容。

课程结束时，学生应能：

• 陈述序列收敛和级数收敛的定义。

• 运用各种定义法以及归纳证明法等标准方法，证明与收敛数列和收敛级数相关的简单命题。

• 根据基本原理并运用极限函数以及级数收敛检验等标准结果，确定不同数列和级数的收敛情况。

• 知道如何将收敛级数的理论运用到幂级数函数的研究中，并计算初等函数的泰勒级数。

Sequences and Series 序列与级数

Credits: 3

This module complements the core Real Analysis module by developing the notion of convergence in discrete contexts. Here the underlying functions are sequences of real numbers, and the fundamental concept of a convergent series is explored in depth.

Significant emphasis is placed on mathematical proof, and the development of fundamental skills and techniques that underpin much of contemporary mathematics.

By the end of the module, students should be able to:

• State the definitions of convergence for both sequences and series.

• Use definitions and standard techniques, such as proof by induction, to construct proofs of simple statements involving convergent sequences and series.

• Determine the convergence of various sequences and series from first principles, and using standard results, such as the algebra of limits and the series convergence tests.

• Understand how the theory of convergent series may be applied to the study of functions defined by power series, and compute Taylor series of elementary functions.