### 数学代写|数论作业代写number theory代考|MATH538101

statistics-lab™ 为您的留学生涯保驾护航 在代写数论number theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数论number theory代写方面经验极为丰富，各种代写数论number theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数论作业代写number theory代考|IRRATIONALITY OF THE EXPONENTIAL CONSTANT

Once we get beyond radical expressions and decimals, irrationality proofs, for the most part, become significantly harder. A notable exception is the irrationality of the exponential constant $e$. Apart from the intrinsic interest of the result, its proof provides our first glimpse of an idea which will recur again and again in irrationality arguments, and which we shall employ extensively in Chapters 2 and $5 .$

Theorem 1.9. The exponential constant $e$ is irrational.
Proof. Assume that $e=p / q$ is rational. That is,
$$\frac{p}{q}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots,$$
and for any positive integer $n$, we have
$$\frac{p n !}{q}=n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\cdots+1+R,$$
where $R$ (which depends on $n$ ) is given by
$$R=\frac{n !}{(n+1) !}+\frac{n !}{(n+2) !}+\cdots$$
We can estimate $R$ in terms of a geometric series:
$$R=\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\cdots<\frac{1}{n+1}+\frac{1}{(n+1)^{2}}+\cdots=\frac{1}{n} .$$
In particular, choose $n=q$. Then
$$R=\frac{p n !}{q}-\left(n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\cdots+1\right)$$
is clearly an integer; but using (1.1), we have $0<R<1$. This is impossible, and so $e$ is irrational.

Observe that this proof relies essentially on an infinite series for $e$, and therefore has to involve concepts of calculus. In some sense this may be surprising, as number theory is usually thought of as studying discrete systems while calculus is the science of the continuous; in another sense there should be no surprise, as it is not even possible to define the number $e$ without recourse to calculus techniques. Whether it is in fact a surprise or not, we shall find that many of our future proofs will be expressed in terms of calculus.

## 数学代写|数论作业代写number theory代考|OTHER RESULTS, AND SOME OPEN QUESTIONS

It is known that $\pi$ is irrational: we shall prove this in the next chapter. It is not hard to see that at least one of the numbers $\pi+e$ and $\pi e$ must be irrational (in fact, at least one must be transcendental – see Chapter 3 ); although, most likely, both are irrational, this has not been proved for either one individually. As a consequence of a difficult result due to Gelfond and Schneider (Theorem $5.18$ ) we know that $e^{\pi}$ is irrational; however it is still unknown whether or not $\pi^{e}$ is irrational. It can also be shown that various numbers such as, for example, $e^{\sqrt{2}}$ and $2^{\sqrt{2}}$ are irrational. However, the irrationality of $\pi^{\sqrt{2}}$ and $2^{e}$, and that of the Euler-Mascheroni constant
$$\gamma=\lim {n \rightarrow \infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\log n\right)=0.57721 \cdots$$ remain undecided. Another problem which has attracted much attention is to investigate the irrationality of the numbers $\zeta(n)$. Here $n \geq 2$ is an integer and $\zeta$ is the Riemann zeta function defined by $$\zeta(s)=\sum{k=1}^{\infty} \frac{1}{k^{s}}=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\cdots$$
for $s>1$. By methods of complex integration we can show that if $n$ is even then $\zeta(n)$ is a rational number times $\pi^{n}$, and this is known to be irrational. On the other hand, it is much harder to find out anything of interest about $\zeta(n)$ for odd $n$. In 1978 the French mathematician R. Apéry sensationally proved that $\zeta(3)$ is irrational. His complicated argument had the appearance of being completely unmotivated, and all of the techniques he had used would have been available two centuries earlier: for these reasons, few people believed that the proof could possibly be correct. Nevertheless it was found possible eventually to confirm all of Apéry’s assertions and thereby establish what has been called “a proof that Euler missed”. A brief (but not easy!) account of Apéry’s work is given in [66].

## 数学代写|数论作业代写number theory代考|IRRATIONALITY OF THE EXPONENTIAL CONSTANT

$$\frac{p}{q}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots$$

$$\frac{p n !}{q}=n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\cdots+1+R$$

$$R=\frac{n !}{(n+1) !}+\frac{n !}{(n+2) !}+\cdots$$

$$R=\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\cdots<\frac{1}{n+1}+\frac{1}{(n+1)^{2}}+\cdots=\frac{1}{n}$$

$$R=\frac{p n !}{q}-\left(n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\cdots+1\right)$$

## 数学代写|数论作业代写number theory代考|OTHER RESULTS, AND SOME OPEN QUESTIONS

$$\gamma=\lim n \rightarrow \infty\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\log n\right)=0.57721 \cdots$$

$$\zeta(s)=\sum k=1^{\infty} \frac{1}{k^{s}}=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\cdots$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。