## 数学代写|实分析作业代写Real analysis代考|KMA321

2022年7月26日

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## 数学代写|实分析作业代写Real analysis代考|Series of Real Numbers

In this section, we will give a brief introduction to series of real numbers. Some knowledge of series, especially series with nonnegative terms, will be required in Chapter 4. The topic of series in general, including various convergence tests, alternating series, etc., will be treated in much greater detail in Chapter

1. We begin with some preliminary notation. If $\left{a_{n}\right}_{n=1}^{\infty}$ is a sequence in $\mathbb{R}$ and if $p, q \in \mathbb{N}$ with $p \leq q$, set
$$\sum_{k=p}^{q} a_{k}=a_{p}+a_{p+1}+\cdots+a_{q}$$
DEFINITION 3.7.1 Let $\left{a_{n}\right}_{n=1}^{\infty}$ be a sequence of real numbers. Let $\left{s_{n}\right}_{n=1}^{\infty}$ be the sequence obtained from $\left{a_{n}\right}$, where for each $n \in \mathbb{N}, s_{n}=$ $\sum_{k=1}^{n} a_{k}$. The sequence $\left{s_{n}\right}$ is called an infinite series, or series, and is denoted either as
$$\sum_{k=1}^{\infty} a_{k} \quad \text { or as } \quad a_{1}+a_{2}+\cdots+a_{n}+\cdots$$
For each $n \in \mathbb{N}, s_{n}$ is called the $n$th partial sum of the series and $a_{n}$ is called the $n$th term of the series.

The series $\sum_{k=1}^{\infty} a_{k}$ converges if and only if the sequence $\left{s_{n}\right}$ of $n$th partial sums converges in $\mathbb{R}$. If $\lim {n \rightarrow \infty} s{n}=s$, then $s$ is called the sum of the series, and we write
$$s=\sum_{k=1}^{\infty} a_{k}$$
If the sequence $\left{s_{n}\right}$ diverges, then the series $\sum_{k=1}^{\infty} a_{k}$ is said to diverge.

## 数学代写|实分析作业代写Real analysis代考|Limits and Continuity

The concept of limit dates back to the late seventeenth century and the work of Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716). Both of these mathematicians are given historical credit for inventing the differential and integral calculus. Although the idea of “limit” occurs in Newton’s work Philosophia Naturalis Principia Mathematica of 1687 , he never expressed the concept algebraically; rather he used the phrase “ultimate ratios of evanescent quantities” to describe the limit process involved in computing the derivatives of functions.

The subject of limits lacked mathematical rigor until 1821 when AugustinLouis Cauchy (1789-1857) published his Cours d’Analyse in which he offered the following definition of limit: “If the successive values attributed to the same variable approach indefinitely a fixed value, such that finally they differ from it by as little as desired, this latter is called the limit of all the others.” Even this statement does not resemble the modern delta-epsilon version of limit given in Section 1. Although Cauchy gave a strictly verbal definition of limit, hee did usé épsilons, déltas, and inéqualities in his proós. For this réason Cauchy is credited for putting calculus on the rigorous basis with which we are familiar today.

Based on the previous study of calculus, the student should have an intuitive notion of what it means for a function to be continuous. This most likely compares to how mathematicians of the eighteenth century perceived a continuous function; namely one that can be expressed by a single formula or equation involving a variable $x$. Mathematicians of this period certainly accepted functions that failed to be continuous at a finite number of points. However, even they might have difficulty envisaging a function that is continuous at every irrational number and discontinuous at every rational number in its domain. Such a function is given in Example $4.2 .2(\mathrm{~g})$. An example of an increasing function having the same properties will also be given in Section 4 of this chapter.

In Section 1 we define the limit at a point of a real-valued function defined on a subset of a metric space, and provide numerous examples to illustrate this idea. In Sections 2 and 3 we consider the closely related theory of continuity and investigate some of the consequences of this very important concept.

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Series of Real Numbers

1. 我们从一些初步的符号开始。如果 Veft{a_{n}\right}_{n=1}^{linfty} $}$ 是一个序列 $\mathbb{R}$ 而 如果 $p, q \in \mathbb{N}$ 和 $p \leq q$ ， 放
$$\sum_{k=p}^{q} a_{k}=a_{p}+a_{p+1}+\cdots+a_{q}$$
定义 3.7.1 让 Vleft{a_{n}\right } } _ { – } { \mathrm { n } = 1 } ^ { \wedge } { \text { {infty} } } \text { 是一个实数序列。让 } Veft{s_{n}\right } { { n = 1 } ^ { \wedge } { \text { linfty} } } \text { 是从获得的序列 Veft } { a _ { – } { n } \backslash r i g h t } \text { , 其中每个 } $n \in \mathbb{N}, s_{n}=\sum_{k=1}^{n} a_{k}$. 序列 $\backslash$ 左{s_{n}\右 $}$ 被称为无限系列或系列，并表示为
$$\sum_{k=1}^{\infty} a_{k} \quad \text { or as } \quad a_{1}+a_{2}+\cdots+a_{n}+\cdots$$
对于每个 $n \in \mathbb{N}, s_{n}$ 被称为 $n$ 该系列的部分总和和 $a_{n}$ 被称为 $n$ 该系列的第 th 项。
该系列 $\sum_{k=1}^{\infty} a_{k}$ 当且仅当序列收敛 咗{5_{n}\右 $}$ 的 $n$ 部分和收敛于 $\mathbb{R}$. 如果
$\lim n \rightarrow \infty s n=s$ ，然后 $s$ 称为级数之和，我们写
$$s=\sum_{k=1}^{\infty} a_{k}$$
如果序列 $\backslash$ 左{5_{n}}右} 发散，然后系列 $\sum_{k=1}^{\infty} a_{k}$ 据说发散。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。